1

The Magnitude-Frequency Distribution of Triggered Earthquakes 1

Stephen Hernandez, Emily E. Brodsky, and Nicholas J. van der Elst, 2

Department of Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064 3

Abstract 4

Large dynamic strains carried by seismic waves are known to trigger seismicity far from 5

their source region. It is unknown, however, whether surface waves trigger only small 6

earthquakes, or whether they can also trigger societally significant earthquakes. To 7

address this question, we use a mixing model approach in which total seismicity is 8

decomposed into 2 broad subclasses: “triggered” events directly initiated or advanced by 9

far-field dynamic stresses, and “untriggered/ambient” events consisting of everything else. 10

The b value of a composite data set is decomposed into a weighted sum of b values of its 11

constituent components, bT and bU. For populations of Californian earthquakes subjected 12

to the largest dynamic strains, the fraction of earthquakes that are likely triggered, fT, is 13

estimated via inter-event time ratios and used to invert for bT. The distribution of bT is 14

found by nonparametric bootstrap resampling of the data, giving a 90% confidence 15

interval for bT of [0.74, 2.79]. This wide interval indicates the data are consistent with a 16

single-parameter Gutenberg-Richter distribution governing the magnitudes of both 17

triggered and untriggered earthquakes. Triggered earthquakes therefore seem just as 18

likely to be societally significant as any other population of earthquakes. 19

Keywords: Gutenberg-Richter, magnitude-frequency distribution, earthquake triggering, R 20

statistic, bootstrap 21

22

2

Introduction 23

Transient strains delivered by large amplitude seismic waves are frequently associated 24

with seismicity rate increases in the far-field [Hill et. al., 1993; Velasco et. al., 2008]. This 25

triggering phenomenon is frequently attributed to dynamic stresses since static stresses decay 26

quickly at such large distances (≥2-3 fault lengths) [King et. al., 1994]. One of several 27

outstanding problems associated with remote dynamic triggering is whether the magnitudes of 28

triggered earthquakes are significantly different from the magnitudes of ambient seismicity. For 29

instance, Parsons and Velasco [2011] investigated whether large (MW≥7) events are capable of 30

dynamically triggering other large (5≤MW≤7) earthquakes in the far-field, and found that they 31

were unable to observe near-instantaneous triggering of large events in the far-field. However, 32

stochastic models of seismicity employ a cascade of small events culminating in large, societally 33

significant earthquakes [Felzer et. al., 2002; Felzer et. al., 2004]. With this prospect in mind, we 34

seek to determine if populations of earthquakes including many remotely triggered events have a 35

different magnitude distribution than those that contain very few triggered earthquakes. 36

This article begins with an overview of earthquake magnitude-frequency distributions, 37

followed by a discussion of a mixing model that relates the observable b-value of a mixed group 38

of triggered and untriggered earthquakes to the parameter of interest (the b-value of the triggered 39

events bT). Utilizing this model requires constructing populations of earthquakes with a known 40

fraction of triggered events. We therefore proceed to form groups of earthquakes that have been 41

affected by dynamic strains of similar amplitude. We can then measure the fraction of triggered 42

events in each of these groups using an inter-event time statistic and proceed to interpret the 43

composite b-value with the aid of a statistical simulation. 44

45

3

Earthquake Magnitude Distributions 46

The magnitude-frequency distribution of earthquakes over broad swaths of regions and 47

time can be represented by the cumulative Gutenberg-Richter (GR) distribution 48

49

log10 (N > M ) = a − b ⋅M (1) 50

51

where a and b are constants and M≥MC, where MC is the magnitude of completeness of the 52

catalog [Ishimoto and Iida, 1939; Gutenberg and Richter, 1942]. The Aki-Utsu maximum 53

likelihood estimator for the parameter b is 54

55

b = log10 (e)〈M 〉 −MC (2) 56

57

where ⟨M⟩ is the mean magnitude [Aki, 1965; Utsu, 1965]. 58

The Gutenberg-Richter distribution is a representation of the exceedance probabilities for 59

a given range of magnitudes. Differences in b-value, therefore, represent differences in the 60

relative hazard of large earthquakes in different regions. Characterizing and identifying 61

differences in b-values has implications for both hazard analysis and the physical mechanisms of 62

earthquake nucleation [Frohlich and Davis, 1993; Utsu, 1999]. 63

Mixing Model and Fraction Triggered 64

Since the maximum likelihood estimate of the b-value is tied to the mean magnitude, the 65

b value of a dataset composed of both triggered and untriggered events can be expressed as 66

67

4

1bmix

= fT ⋅1bT

+ (1− fT ) ⋅1bU (3) 68

69

where bmix, bT, bU are the b values of the mixed (composite), triggered, and untriggered events 70

and fT is the estimated fraction of the mixed dataset consisting of triggered earthquakes 71

(Appendix A). To use equation (3), we need to construct populations of earthquakes with 72

differing fractions triggered and then perform two distinct tasks: (1) measure bmix in the 73

combined population and (2) determine the fraction of triggered events. Additionally, we need 74

to find a group of earthquakes with a very low fraction of triggering in order to estimate the 75

untriggered b-value bU. 76

We will accomplish all of these goals by capitalizing on the previous observation that the 77

fraction of triggered events in the far-field is a well-defined function of the peak amplitude of the 78

seismic waves, i.e., larger amplitude waves trigger more events [van der Elst and Brodsky, 79

2010]. Therefore, we can construct groups of earthquakes which follow dynamic strains from 80

distant earthquakes. The groups of earthquakes following large amplitude shaking will have a 81

large (and measurable) triggered fraction and can be used in conjunction with equation (3) to 82

measure the b-value of triggered earthquakes, bT. Those following small or extremely distant 83

earthquakes will have a very low triggered fraction and can be used to approximate bU. Note that 84

this definition of the untriggered population may include many earthquakes that are triggered by 85

other, unidentified local mainshocks. It has been proposed elsewhere that the fraction of locally 86

triggered events catalog-wide is >80% and so essentially any group of earthquakes will contain 87

aftershocks [Marsan et. al., 2008]. However, for the purpose of this study we are asking if a 88

group of identifiable, remotely triggered events has magnitude behavior that is distinct from 89

5

other groups of earthquakes. This is a key question for both operational forecasting and physical 90

understanding of the dynamic triggering process. 91

Data and Analysis Method 92

Our data consist of source parameters accessed from the Advanced National Seismic 93

Network (ANSS) composite catalog for Californian and Nevadan seismic networks (32o to 42oN, 94

124o to 114oW). Event location, depth, origin time, and preferred magnitude are extracted for 95

1984 - 2011. Events with depth greater than 30 km and magnitude less than M2.0 are discarded 96

to yield a complete, uniform catalog. The b value and 95% confidence level (as prescribed by Aki 97

[1965]) for the composite catalog are 1.019 ± 0.005. 98

Our local target catalog is partitioned into grids of size 0.1°x0.1° . For each node, we 99

loop through a global catalog of 'test triggers.' Events with magnitude MW>5 and origin time after 100

1984 qualify for inclusion as a test trigger. Depths are limited to those shallower than 50 km 101

because of their greater relative efficiency at generating surface waves. Additionally, each 102

global test trigger must be at least 400 km from the centroid of the local bin. 103

In order to identify earthquake populations with strong triggering, we need to estimate 104

the local peak ground velocity from a distant earthquake. We make this estimate by inverting the 105

standard empirical magnitude regression for surface waves 106

107

log10 (A) = MS −1.66 ⋅ log10 (Δ)− 2 (4) 108

109

where A is the peak amplitude (in microns) for the 20-second Rayleigh surface wave and Δ is the 110

epicentral distance in degrees [Lay and Wallace, 1994]. We estimate strain, ε, by assuming ε = 111

V/CS, where V is the particle velocity (approximately equal to 2·π·f·A [Aki and Richards, 2002]) 112

6

and CS is the Raleigh wave phase velocity estimated at 3.5 x 109 µm/s. Second order effects due 113

to faulting style and subsequent radiation pattern can result in errors as high ±20% for individual 114

events [Gomberg and Agnew, 1996], but the global average curve will accurately predict the 115

average behavior of a large group of potential triggering events. 116

For a group of target areas that have all experienced a given amplitude of shaking we can 117

measure the inter-event time ratio, or R, as a tool to measure the fractional rate change [van der 118

Elst and Brodsky, 2010]. The time ratio is defined as 119

120

R ≡ t2t1 + t2 (5) 121

122

where t1 and t2 are, respectively, the times to the first earthquake before (with magnitude M1) and 123

the first earthquake after (with magnitude M2) the arrival of seismic energy from some potential 124

far-field trigger (Figure 1). In the presence of triggering, there will be a slight bias toward small 125

values of t2, leading to a larger proportion of small-valued R-ratios and a mean value of R less 126

than 0.5. 127

Prior work has shown that measurements of the mean value of R can be used to estimate 128

rate changes induced by static stresses near a mainshock [Felzer and Brodsky, 2005] or changes 129

in response to dynamic stresses like the passage of transient surface waves [van der Elst and 130

Brodsky, 2010]. In particular, 131

132

〈R〉 = 1δλ 2 ⋅[(δλ +1) ⋅ ln(δλ +1)-δλ]

(6) 133

134

7

where δλ = (λ2 - λ1)/λ1, and λ1 and λ2 are the rates of seismicity before and after the arrival of the 135

purported trigger [van der Elst and Brodsky, 2010, equation (2)]. For the purpose of utilizing 136

equation (3), we need to measure the fraction of a dataset comprised by triggered earthquakes, 137

i.e., 138

fT =

δλδλ +1 (7) 139

where δλ is estimated from measured R ratios (Appendix B). 140

The finite length of an earthquake catalog introduces a bias in the calculation of R values. 141

Unequal conditioning of t1s and t2s (Figure 1) can be introduced as the test trigger time moves 142

away from the middle of the local catalog. We remove this bias by truncating our local catalog to 143

windows symmetric about the arrival time of each test trigger. The analysis is run for symmetric 144

windows of fixed lengths ±1, 2, and 4 years, and also for symmetric windows of varying lengths. 145

The choice of window length has a significant effect on the number of samples generated overall 146

. For example, a ±1 year window applied to grid nodes with very low seismicity rates (e.g., the 147

San Joaquin Valley in Central California) will exclude R-ratios that a longer window (e.g., ±2 or 148

±4 years) would not. Conversely, an exceedingly large window has the potential of excluding R 149

ratio calculations using data near the beginning and end of the catalog. Since the choice of 150

window length does not significantly alter the conclusions of the study, we report in the main 151

text results generated by imposing a 2 year symmetric window as it maximizes the number of 152

samples. Additional time window results are in the supplementary materials. 153

Once fT has been measured for each group of earthquakes with similar values of dynamic 154

strain, we then measure the observable, composite b-value (bmix) and proceed to infer a range of 155

values for bT that is consistent with the data. 156

157

8

Results 158

Our primary objective is to evaluate the change in b-value, or mean magnitude, of 159

earthquakes relative to imposed strain. However, as discussed above, we first need to establish 160

measurements of the fraction triggered for our selected dataset. As anticipated, the triggering 161

intensity increases monotonically for strains larger than ~10-7 (Figure S2). This reaffirms the 162

results of van der Elst and Brodsky [2010] with a different method to eliminate the finite catalog 163

bias than was originally used. 164

We now proceed to explore the magnitude distribution of each dataset. Figure 2 shows 165

the Aki-Utsu b-value as a function of peak dynamic strain. The b-values for the M1 and M2 166

populations of earthquakes are plotted along with their 95% confidence intervals based on 167

10,000 bootstraps [Efron and Tibshirani, 1994]. In general, the b-values for either population do 168

not show significant variations with dynamic strains. These results suggest that increased 169

triggering does not significantly perturb the magnitude distribution. We now have measurements 170

of bmix. 171

The final requirement to infer bT is to measure bU (our reference background seismicity 172

parameter). We initially attempted to define bU using events from the population of “M1” events. 173

In that case, we supposed that the magnitude of the event immediately preceding the arrival of 174

dynamic stress was an accurate representation of the steady-state distribution in a system 175

unperturbed by far-field transient waves. 176

As can be seen in Figure 2, it appears using only the M1s to measure b biases the 177

measured values above the catalog average b-value of 1.019 ± 0.005. This sampling bias arises 178

from the definition of R. As seismicity rates are high after local mainshocks, inter-event times 179

are low. Therefore, distant earthquakes are unlikely to occur soon enough after a large 180

9

magnitude, local mainshock to include these events in the estimation of the pre-trigger 181

magnitudes (M1s). The local aftershocks effectively shield the larger magnitude mainshock from 182

being sampled as an example of the pre-trigger seismicity by the R-statistic. As a result, the b-183

value inferred solely from the M1s is systematically higher (lower mean magnitude) than the b-184

value of the M2s. A similar shielding effect exists from foreshock activity for the M2s and 185

therefore both the M1- and M2-inferred b-values are systematically higher (mean magnitudes are 186

lower) than the catalog-averaged b-value with systematic offset between them, even for the very 187

low perturbing strains (For details see Appendix C). 188

We therefore need to incorporate this bias into our comparisons by only comparing 189

similarly conditioned magnitudes. Since the M2 group of magnitudes will contain the highest 190

number of triggered events, the most meaningful reference magnitudes must also come from the 191

M2 data set. We take the values of M2 from the lowest strain dataset (which contains the smallest 192

triggered fraction) as the value of bU. 193

Next we apply the mixing model to data from the ANSS catalog to extract bT from the 3 194

observable variables: bmix, bU, and fT. As an initial example, the triggered fraction and bmix are 195

estimated from the R-ratios and M2 data associated with the largest 314 strains in the dataset. The 196

largest 314 strains are used because they correspond to strain measurements, ε, greater than 10-6 197

(the last bin in Figure 2). A reference b-value, bU, against which we will compare estimates of bT, 198

must also be estimated. We choose data from the M2 magnitudes associated with the 105 smallest 199

strains ( ε < ~1.25 x10-9, equivalent to the data in the first bin of M2 data in Figure 2) to represent 200

the parameter bU. 201

Table 1 summarizes the data for all components of the mixing model for the Northern 202

California-Nevada (Lat. > 37°), Southern California-Nevada (Lat. < 37°), and full data sets (32° 203

10

< Lat. < 42°). We focus here on results from Northern California-Nevada, i.e., latitudes > 37°, 204

because these data contain a particularly robust triggering signal. We obtain bmix = 1.26, bU = 205

1.11, and a value of fT indicating ~25% of the 314 events are triggered earthquakes, where bmix 206

and bU are systematically higher than the catalog-averaged b = 1.019 ± 0.005 because of 207

sampling biases (Appendix C). The bT associated with these data and inverted from the mixing 208

model is 2.03. At face value, this observation suggests that b-values of data with high percentage 209

of triggered earthquakes are on average larger than the b-values of untriggered/ambient 210

seismicity and is consistent with prior results from previous studies addressing this same 211

question [Parsons and Velasco, 2011; Parsons et. al., 2012]. 212

To test the significance of these results, we employ the statistics of nonparametric 213

bootstrap resampling. Suppose fT* and bmix* indicate a size Nboot vector of bootstrap resamplings 214

(with replacement) of the original data, R-ratios and M2s, respectively. Then, for some arbitrarily 215

large value of Nboot, the distribution of bT can be approximated by bT*, where each bTi* is 216

inverted from equation (3) as before. Furthermore, we partition the Northern California-Nevada 217

data set into four discrete amplitude bins, each with 500,000 bootstrap iterations, to make four 218

independent estimates of the distribution of bT (Figure 3). The individual curves are then 219

combined to generate a marginal distribution of bT (Figure 4). The resulting values confirm that 220

the magnitude distribution of the remotely triggered earthquakes is indistinguishable from the 221

untriggered population. 222

We also use a synthetic catalog of magnitudes to test whether our analysis is sensitive 223

enough to resolve when bT is genuinely different from bU. We produce the synthetics mimicking 224

the data in each amplitude range of Figure 3 by using a Monte Carlo simulation to produce 225

magnitudes following a Gutenberg-Richter distribution with a prescribed value of bT for the 226

11

triggered fraction (fT) of the group and a distinct value of bU for the untriggered (1-fT) fraction. 227

The total number of events in each group matches the observed number and fT is set to match the 228

observed mean value. We then invert the synthetics for bT. In order to reproduce the 229

observational uncertainty of fT, for each group we draw an inferred value of fT from the observed 230

empirical distribution of fT and then invert using a fixed value of bU and Eq. 3. This procedure is 231

repeated a total of 500,000 times for each amplitude range of interest. Those individual curves 232

for bT are then stacked and weighted to produce a marginal distribution of bT as was done in 233

Figure 4. We find that for prescribed values of bT > 1.5, the distribution of synthetic triggered 234

magnitudes is markedly different from the marginal distribution generated from observable data 235

(Figure S4). This lends credence to the methodology’s ability to resolve differences between bT 236

and bU should they exist. 237

Implications 238

Our results indicate there are no discernible differences between the magnitude 239

distributions of data sets that likely contain a high proportion of triggered events, versus data sets 240

that do not contain high proportions of triggered events. The observed composite b values show 241

no significant variation with respect to calculated peak dynamic strains (Figure 2). The small 242

variations that do exist do not show a monotonic increase with the fraction of likely triggered 243

events in the population, meaning that this conclusion is not likely to change with increased 244

statistical resolution. Inverting for bT directly using a mixing model does not alter these 245

conclusions (Figure 4). 246

These interpretations are significantly different than that of Parsons and Velasco [2011] 247

where it is suggested that dynamically triggered earthquakes are preferentially small. These 248

studies differ in several fundamental ways. For instance, the statistical treatment here includes 249

12

2074 examples of potential triggers, including some very weak triggers, and deals explicitly with 250

the fact that any observed group of earthquakes is a mixture of triggered and untriggered events 251

via our mixing model. The previous work had only 205 examples of very strong triggering. 252

While this paper’s approach includes a larger number of cases, the potential exists for 253

contamination from background seismicity. Secondly, the R-statistic uses an optimized, adaptive 254

time window to measure rate changes and therefore is inherently more sensitive than the 255

counting method of Parsons and Velasco. Finally, we focus on small earthquakes as diagnostic of 256

the magnitude distribution; this contrasts with the approach of Parsons and Velasco where only 5 257

< MW < 7 events are examined. The advantage of this paper’s approach is that larger numbers 258

ensure statistical robustness since larger magnitude earthquakes are intrinsically rare and may not 259

be expected to be observed during a short time interval. However, if larger magnitude 260

earthquakes have distinct physical nucleation mechanisms our approach has no bearing on them. 261

Thus far, there is little direct evidence of a break-down in earthquake self-similarity, therefore a 262

separate process governing large earthquakes could be a relatively unlikely possibility, but one 263

that must remain open nonetheless. 264

Conclusions 265

We find that the statistical evidence for fundamentally different underlying distributions 266

between triggered and untriggered earthquakes is weak. This strongly supports the idea that the 267

magnitudes of triggered and untriggered earthquakes are randomly drawn from a single 268

parameter GR distribution, at least for moderate magnitudes. Therefore, remotely triggered 269

earthquakes are likely to be as large as any other group of seismicity. However, since the total 270

number of triggered events is small, the probability of observing a remotely triggered large 271

earthquake is accordingly small. In summary, we have opted for large numbers in our statistical 272

13

treatment and arrived at conclusions distinct from prior efforts. This suggests that the question of 273

the magnitude distribution of dynamically triggered events is still an open problem. 274

References 275

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confidence limits, Bull. Earthquake Res. Inst. Tokyo Univ., 43, 237–239. 277

Aki, K., and P. G. Richards (2002), Quantitative Seismology, 2nd ed., 700 pp., Univ. Sci. Books, 278

Sausalito, Calif. 279

Brodsky, E. E. (2006), Long-range triggered earthquakes that continue after the wave train 280

passes, Geophys. Res. Lett., 33, L15313, doi:10.1029/2006GL026605. 281

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Felzer, K. R., and E. E. Brodsky (2005), Testing the stress shadow hypothesis, J.Geophys. Res., 284

110, B05S09, doi:10.1029/2004JB003277. 285

Felzer, K. R., T. W. Becker, R. E. Abercrombie, G. Ekstrom, and J. R. Rice (2002), Triggering 286

of the 1999 MW 7.1 Hector Mine earthquake by aftershocks of the 1992 MW 7.3 287

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Felzer, K. R., R. E. Abercrombie, and G. Ekstrom (2004), A common origin for aftershocks, 289

foreshocks, and multiplets, Bull. Seismol. Soc. Am., 94, 88–98. 290

Frohlich, C., and S. D. Davis (1993), Teleseismic b values; or much ado about 1.0, J. Geophys. 291

Res., 98 (B1), 631–644, doi:10.1029/92JB01891. 292

Gomberg, J., and D. Agnew (1996), The accuracy of seismic estimates of dynamic strains: An 293

evaluation using strainmeter and seismometer data from Pinon Flat Observatory, 294

California, Bull. Seismol. Soc. Am., 86, 212–220. 295

14

Gutenberg, B., and C. F. Richter (1944), Frequency of earthquakes in California, Bull. Seismol. 296

Soc. Am., 4, 185–188. 297

Hill, D. P., et al. (1993), Seismicity remotely triggered by the magnitude 7.3 Landers, California, 298

earthquake, Science, 260(5114), 1617–1623. 299

Ishimoto, M., and K. Iida (1939), Observations of earthquakes registered with the 300

microseismograph constructed recently, Bull. Earthquake Res. Inst. Univ. Tokyo, 17, 301

443–478. 302

King, G. C. P., R. S. Stein, and J. Lin (1994), Static stress changes and the triggering of 303

earthquakes, Bull. Seismol. Soc. Am., 84, 935–953. 304

Lay, T., and T. C. Wallace (1995), Modern Global Seismology, Academic, San Diego, Calif. 305

Marsan, D., and O. Lengliné (2008), Extending earthquakes' reach through cascading, Science, 306

319, 1076–1079. 307

Parsons, T., and A. A. Velasco (2011), Absence of remotely triggered large earthquakes beyond 308

the mainshock region, Nature Geoscience, 4, 321–316, doi:10.1038/ngeo1110. 309

Parsons, T., J. O. Kaven, A. A. Velasco, and H. Gonzalez-Huizar (2012), Unraveling the 310

apparent magnitude threshold of remote earthquake triggering using full wavefield 311

surface wave simulation, Geochem. Geophys. Geosyst., 13, Q06016, 312

doi:10.1029/2012GC004164. 313

van der Elst, N. J., and E. E. Brodsky (2010), Connecting near-field and far-field earthquake 314

triggering to dynamic strain, J. Geophys. Res., 115, B07311, doi:10.1029/2009JB006681. 315

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15

Utsu, T. (1965). A method for determining the value of b in the formula log n = a - bM showing 318

the magnitude-frequency relation for earthquakes, Geophys. Bull. Hokkaido Univ., 13, 319

99–103 (in Japanese with English summary). 320

321

Figure 1: Schematic cartoon of hypothetical synthetic data from a single grid node and 322

demonstrating the procedure for measuring R ratios. The vertical black dashed line is the 323

approximate arrival time of seismic energy from one far-field event, i.e., ‘test trigger’. For each 324

test trigger, t1, t2, and M2 are the primary variables used to assess triggering intensity (δλ) and 325

magnitude distribution of triggered earthquakes over a broad region. In this hypothetical 326

example, a symmetric window of ±1 day (24 hours) is imposed. In practice, we impose a 327

symmetric window of ±2 years. 328

Table 1: Data summary for largest amplitude bin (ε > 1x10-6)

Northern California-Nevada, N=314

Southern California-Nevada, N=141

Full ANSS, N=455

bmix 1.26 1.32 1.28

bU 1.11 1.06 1.08

δλ 0.34 -0.11 0.18

fT 0.25 Undef.a 0.15

bT 2.03 Undefa. 273.99b

a: The mixing model requires that bT be a positive quantity which is not possible if an overall negative rate change is observed. b: The model outputs an unrealistic large value of bT due to the low overall fraction of triggered events and the large value of bmix relative to bU. For this reason, we focus on the Northern California data with a large triggering signal.

16

Figure 2: b-value for the Northern California-Nevada target catalog as a function of peak 329

dynamic strain and for a fixed symmetric window of ±2 years to correct for finite catalog effects. 330

Each decade is evenly binned into thirds. The b-value of the M2s, a proxy for the mean 331

magnitude of the dataset, is consistently lower (corresponding to higher mean magnitude) than 332

the b value for the M1s, due to a sampling bias (Appendix C). The variances for the estimates 333

associated with the largest strain are largest because of the small sample size. 334

Figure 3: Distributions of bT as inverted from distinct amplitude (strain) bins. Data come from 335

the Northern California-Nevada data set. Four disjoint datasets with amplitudes ranges i)10-7- 336

2.15x10-7 (N=4780), ii.) 2.15x10-7 - 4.64x10-7 (N=1974), iii.) 4.64x10-7 - 10-6 (N=905), iv.) >10-6 337

(N=314) are constructed. We treat each disjoint subset as independent measurements of fT and 338

bmix. A distribution for bT is then estimated after generating 500,000 bootstraps of the data. The 339

data are smoothed via Gaussian kernel density estimation. 340

Figure 4: The distributions of bU and the marginal distribution of bT, generated via bootstrap 341

resampling The distribution of bT is a weighted sum of densities generated from individual 342

amplitude bins (Figure 3), i.e. Densitystack = fTiniNDensityi

i=1

Namplitudes

∑ where Namplitudes is the number 343

of amplitudes bins, ni is the number of measurements in bin i, fT is the mean estimate of the 344

fraction triggered in bin i, Densityi is the probability density estimate of bT using data from bin i 345

and N is the total number of measurements. The 90% confidence interval (vertical red dashed 346

lines) spans a wide margin and the distribution for bT (blue curve) cannot be distinguished from 347

our preferred distribution for bU (green curve). bT is heavy-tailed toward large values of bT. 348

!2 !1.5 !1 !0.5 0 0.5 1 1.5 2

2

2.5

3

3.5

4

Magn

itude

Hours since shaking

UntriggeredTriggered

t1 t2

M1

M2

!20 !15 !10 !5 0 5 10 15 20

2

2.5

3

3.5

4

Magn

itude

Hours since shaking

UntriggeredTriggered

FIGURE 1

10 !910 !8

10 !710 !6

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Peak Dynamic Strain

bmix value

M1

M2

FIGU

RE 2

00.5

11.5

22.5

33.5

44.5

50

0.2

0.4

0.6

0.8 1

1.2

1.4

1.6

1.8 2

10!07 to 2.15x10

!07

2.15x10!07 to 4.64x10

!07

4.64x10!07 to 10

!06> 10!06

Density

bT

FIGURE 3

00.5

11.5

22.5

33.5

44.5

50

0.5 1

1.5

b!value

Density

00.5

11.5

22.5

33.5

44.5

5 0 50 100FIG

URE 4

1

Supplemental Materials 1

Appendix A: Mixing Model 2

Suppose a sequence of earthquake magnitudes, Mimix, exists such that it is composed 3

entirely of either triggered, MjT, or untriggered, Mk

U, events. The sum of magnitudes of the mixed 4

(composite) catalog can be expressed as 5

6

(A1) 7

8

where nT and nU are the total number of triggered and untriggered observations, and ntot = nT + 9

nU. Equation (A1) can be recast as a weighted sum of the means of the individual components 10

11

(A2) 12

13

where fT = nT/ntot. Finally, since the Aki-Utsu equation (equation (2) of main text) relates the b-14

value of a given dataset to the mean magnitude of that dataset, it is trivial to show that 15

16

(A3) 17

18

In this formulation, we are assuming that the minimum magnitude of completeness is equivalent 19

for both subcatalogs, MT and MU. Equation (A3) is the same as our mixing model equation in the 20

Mimix

i

ntot

∑ = M jT

j

nT

∑ + MkU

k

nU

∑

Mmix = fT ⋅ MT + (1− fT ) ⋅ MU

1bmix

= fT ⋅1bT

+ (1− fT ) ⋅1bU

2

main text. Application of a maximum likelihood methodology yields similar results [Kijko and 21

Smit, 2012; Appendix A]. 22

Appendix B: Estimating fT 23

Inverting for bT from the mixing model equation derived in Appendix A requires an 24

estimate of the fraction of the data attributed to triggered earthquakes, or fT. For a given source 25

volume with a distribution of faults, we model the seismicity of the volume as a Poisson process 26

with an average intensity parameter λ. Dynamic strains traveling through a source volume can 27

activate faults and induce a step change in the intensity parameter from λ1 to λ2. For a stepwise 28

homogenous Poisson process, we define the fractional rate change induced by some far-field 29

trigger as 30

31

(B1) 32

33

Over one pretrigger recurrence interval (time τ = λ1-1), the expected number of earthquakes in the 34

posttrigger aftermath is nafter = δλ+1. The expected number of earthquakes in the time interval of 35

length τ prior to the putative trigger is nbefore = λ1· λ1-1=1. We can therefore define fT as 36

37

(B2) 38

39

The mean of a sample of R-ratios is used to estimate δλ (via equation (6) of the main text) and fT. 40

Appendix C: Sampling Biases 41

Aftershock Shielding Effect 42

δλ = λ2 − λ1λ1

= λ2λ1

−1

fT =nTntot

= ntot − nUntot

=nafter − nbefore

nafter= (δλ +1)−1

δλ +1= δλδλ +1

3

Throughout the entire range of strains, the M1 population shows a consistently higher b-43

value (corresponding to a lower mean magnitude) than its equivalent population of M2 44

magnitudes (Figure 2). Multiple strategies for correcting the finite catalog effect do not alter this 45

general observation (Figure S1). This shielding effect likely stems from the increase in rate 46

during aftershock sequences. Figure S3 shows a schematic diagram demonstrating the origin of 47

the systematic offset of magnitudes between the M1s and M2s. The magnitudes of strong local 48

mainshocks are generally higher than the bulk of the seismicity immediately preceding these 49

mainshocks. Consistent with Omori’s Law, the time to the next earthquake after a mainshock 50

will, on average, be shorter than the time to the first earthquake that preceded it. As dynamic 51

waves from far-field ruptures periodically perturb local source volumes, they may arrive in the 52

vicinity of the timing of some local mainshock. If so, these far-field dynamic waves are 53

accordingly less likely to fall within the interval between the timing of the mainshock and the 54

mainshock’s first aftershock, versus arriving within the interval between the timing of the 55

mainshock and the first earthquake that precedes it; a large magnitude, local mainshock is 56

systematically biased to be labeled an `M2’ versus being designated as an `M1.’ In other words, 57

the large mainshock is shielded from being labeled an M1 by ensuing aftershocks. 58

Foreshock Shielding Effect 59

The presence of foreshocks induces a similar shadowing effect. Figure 2 in the main text 60

shows that the b-values of the M2s (~1.1) are systematically larger than the b-value of the overall 61

catalog (~1.0). Since foreshock sequences follow an Inverse Omori Law, as far-field triggers 62

approach the time of a local mainshock, they will be immersed in the rate increases (in an 63

average sense) preceding the local mainshock. As the forehocks are less abundant than the 64

4

aftershocks, the shielding effect is less severe and therefore the M2s do not show a significant 65

perturbation from catalog averaged values. 66

ETAS Simulation 67

We can reproduce both of these shielding effects in an epidemic-type aftershock model 68

(ETAS) [Ogata, 1998]. We produced a synthetic catalog following the procedure of Brodsky 69

[2011] and measured the magnitude distribution of the events prior and after randomly selected 70

times. We found for 100 simulations that the M1 b-value = 1.22±0.04 and the M2 b-value = 71

1.0±0.03, with 1 standard deviation reported. 72

As discussed in Brodsky [2011], a standard ETAS simulation has too few foreshocks 73

(relative to aftershocks) compared to the observations. This disparity is due to either 74

completeness problems or a physical propensity for foreshocks. The overabundance of 75

foreshocks is an interesting issue in itself, however, here we are only concerned with its effect on 76

the sampling of magnitudes. Therefore, the standard ETAS simulation only reproduced the 77

aftershock shielding and did not explain the deviation of the M1s in the observations. We 78

therefore performed an additional set of modified ETAS simulations where 25% of the 79

aftershocks were randomly assigned to occur as foreshocks. In this case, the observed aftershock 80

to foreshock ratio was close to the catalog values (~2) and the observed M1 b-value = 1.19±0.03 81

(1 std.) and the M2 b-value = 1.10±0.04 (1 std.), as required by the catalog. Therefore, we 82

conclude that all of the trends visible in Figure 2 of the main text are explained as selection 83

issues and there is no observable distinction between triggered and non-triggered groups. 84

85

5

Auxiliary References 86

Brodsky, E. E. (2011), The spatial density of foreshocks, Geophys. Res. Lett., 38, L10305, 87

doi:10.1029/2011GL047253. 88

Kijko, A. and A. Smit (2012), Extension of the Aki-Utsu b-Value estimator for incomplete 89

catalogs, Bull. Seismol. Soc. Am., 102, 1283–1287. 90

Ogata, Y. (1998), Space-time point-process models for earthquake occurrences, Ann. Inst. Stat. 91

Math., 50 (2), 379–402. 92

Figure S1: b-value for the Northern California-Nevada target catalog as a function of peak 93

dynamic strain. Each decade is evenly binned into thirds. The y-axes are the same for each 94

subplot. (a), b-values from magnitudes (M1 and M2) corresponding to R-ratios calculated with a 95

±1 year symmetric window; (b), as in (a), corresponding to 2-year symmetric windows; (c), as in 96

(a), corresponding to 4-year symmetric windows; (d), as in (a), corresponding to symmetric 97

windows of variable length. In variable symmetry, window lengths will depend on the trigger 98

time’s distance from the beginning or end of the catalog, i.e., ± min{T1, T2}, where T1 = t0 - tBEG 99

(tBEG = 01/01/1984), and T2 = tEND - t0 (tEND = 12/31/2010). For example, if a far-field stress 100

arrives at a particular spatial grid at t0 =January 30, 1984, then min{T1,T2} = 30 days. In order 101

Table S1: Data summary for Nor thern California-Nevada, with different amplitude bins

1x10-7 < ε < 2.15x10-6

N=4780 2.15x10-7 < ε < 4.64x10-6

N=1974 4.64x10-7 < ε < 1x10-6

N=905 ε < 1x10-6

N=314

bmix 1.10 1.16 1.14 1.26

bU 1.11 1.11 1.11 1.11

fT 0.06 0.13 0.15 0.25

bT 0.98 1.65 1.27 2.03

6

to maintain symmetry and to remove the finite catalog bias, all seismicity not within 30 days of t0 102

is ignored. In all cases, the b-value of the M2s, a proxy for the mean magnitude of the dataset, is 103

consistently lower (corresponding to higher mean magnitude) than the b-values for the M1s. 104

These results are consistent even when applying symmetric windows of differing lengths. 105

Figure S2: Triggering intensity (rate change) for the Northern California-Nevada (Lat.≥37°) 106

target catalog as a function of peak dynamic strain. Each decade is evenly binned into thirds. 107

Triggering intensity calculated from R ratios ± 1, 2, and 4 years from the arrival of the putative 108

trigger as well as variable time windows as explained in the caption of Figure S1. Gaps 109

correspond to negative values that have been excluded after a logarithmic transformation was 110

applied. 111

Figure S3: Schematic diagram for the origin of the systematic offset of magnitudes between the 112

M1s and M2s. Teleseismic waves arriving in our target catalog are represented by black arrows. 113

For each far field event, a unique R-ratio is calculated and an M1 and M2 within the local catalog 114

are identified. When a local mainshock occurs, the time to the next event is, on average, 115

substantially reduced as aftershocks tend to cluster in time and space. It is therefore unlikely that 116

teleseismic surface waves will arrive in the time interval between the local mainshock and its 117

first local aftershock. Therefore, the large local mainshocks in a target catalog are rarely 118

designated as M1s, and almost always designated M2s. Hence, the systematic offset in 119

magnitudes between the two datasets. We call this the aftershock shielding effect. 120

Figure S4: Marginal distributions of bT using synthetically derived magnitude sequences. Four 121

tests were conducted, each assuming a different value of bT: bT =1.25, bT =1.5, bT =1.75, bT =2.0 122

(peach, pink, teal, and black). Each curve is the sum of 4 amplitude bins as in figure 3 in the 123

main text. Within each amplitude bin, the requisite number of triggered earthquake are 124

7

generated, then contaminated with magnitudes drawn from a distribution with a different b-value 125

(bU = 1.11). This process is repeated 500,000 times and a density estimate is generated using a 126

Gaussian kernel. The resulting 4 density estimates for bT are then stacked to give a final curve. 127

The only free parameter is the input bT. Thick blue curve are results from observable data within 128

the Northern California-Nevada region. 129

130

−10

−9−8

−7−6

−50.

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FIG

URE

S21

10−9

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10−7

10−6

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10−2

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100

Peak Dynam

ic Strain

Fractional Rate Change

FIGU

RE S2

1−year Symm

etric Window

2−year Symm

etric Window

4−year Symm

etric Window

Variable Symm

etric Window

M1

M1

M2

M2

M2

M1

Far-!eldstrainarrives

Large local event falls into M2

population multiple tim

es...N

ever into the M1 population

Robust aftershock sequence,elevated seism

icity rates

Magntiude

Time

}FIG

URE S3

00.5

11.5

22.5

33.5

44.5

50 0.2

0.4

0.6

0.8 1 1.2

1.4

b!value

Density

bT = 1.25bT = 1.50bT = 1.75bT = 2.00Observation

FIGU

RE S4