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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
Aki, K. (1965), Maximum likelihood estimate of b in the formula log N = a − bM and its 276
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
Efron, B and R.J. Tibshirani (1994), An Introduction to the Bootstrap, 436 pp., Monographs on 282
Statistics and Applied Probability, CRC Press, Boca Raton, FL. 283
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
Landers earthquake, J. Geophys. Res., 107 (B9), 2190, doi:10.1029/2001JB000911. 288
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
Velasco, A. A., S. Hernandez, T. Parsons, and K. Pankow (2008), Global ubiquity of dynamic 316
earthquake triggering, Nature Geoscience, 1, 375–379. 317
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
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.
2
0.4
0.6
0.81
1.2
1.4
1.6
log 10
(Pea
k D
ynam
ic S
train
)
b value + 95% C.I.1
year
sym
met
ric w
indo
w, b
M V
s. S
train
M1
M2
−10
−9−8
−7−6
−50.
2
0.4
0.6
0.81
1.2
1.4
1.6
1.8
log 10
(Pea
k D
ynam
ic S
train
)
b value + 95% C.I.
2 ye
ar s
ymm
etric
win
dow
, bM
Vs.
Stra
in
M1
M2
−10
−9−8
−7−6
−50.
4
0.6
0.81
1.2
1.4
1.6
1.8
log 10
(Pea
k D
ynam
ic S
train
)
b value + 95% C.I.
4 ye
ar s
ymm
etric
win
dow
, bM
Vs.
Stra
in
M1
M2
−10
−9−8
−7−6
−5
0.7
0.8
0.91
1.1
1.2
1.3
1.4
log 10
(Pea
k D
ynam
ic S
train
)
b value + 95% C.I.
Varia
ble
sym
met
ric w
indo
w, b
M V
s. S
train
M1
M2
FIG
URE
S21
10−9
10−8
10−7
10−6
10−3
10−2
10−1
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
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