Embed Size (px)
Citation preview
1
Are spontaneous earthquakes stationary in California? 1
Qi Wang, David. D. Jackson and Jiancang Zhuang 2
Abstract�3
Aftershocks and some main shocks are triggered, with timing controlled by preceding 4
events. The remaining spontaneous earthquakes presumably respond to tectonic 5
stresses. We consider whether triggered events can be reliably identified, whether the 6
rest are stationary, and whether external phenomena control them. To all three 7
questions, some studies of earthquake physics and hazard assume answers. Many 8
suggest that stress changes from large distant earthquakes can alter the local 9
spontaneous earthquake rate. We demonstrate significant differences in the apparent 10
earthquake rate after declustering by different methods and present criteria for 11
assessing the influence of such distant events. The estimated spontaneous earthquake 12
rate depends on both the lower magnitude threshold of included events, whether 13
spontaneity is treated in binary or probabilistic form, as well as assumptions about 14
catalog completeness. Different statistical tests give different answers to the question 15
of stationarity. We examine a reported rate change in southern California and the 16
suggestion that it might have resulted from the 1960 Chile and 1964 Alaska 17
earthquakes. The rate change itself is questionable. If it has occurred, it was probably 18
not caused by those distant events because the rate change is not present in northern 19
California or in all parts of southern California.20
1.�Introduction�21
Attempts to relate earthquake occurrence to external phenomena like tectonic strain 22
rate variations, distant earthquakes, or tidal stresses are confounded by aftershocks 23
and other locally triggered earthquakes. Those earthquakes are numerous, and their 24
timing is dominated by the previous events that triggered them. Thus seismologists 25
have developed procedures for “declustering” aftershock catalogs [e.g., Utsu, 1969; 26
2
Gardner and Knopoff, 1974; Reasenberg, 1985; Kagan et al., 1991; Jones and 27
Hauksson, 1997, Zhuang et al., 2002] leaving a set of earthquakes presumably 28
independent of the local interactions. We ask here whether such declustered catalogs 29
can be used to identify time varying external phenomena that affect the rate of 30
earthquake occurrence. 31
We examine results from different procedures by Reasenberg [1985], Kagan et al.32
[1991, 2006], and Zhuang et al. [2005] for identifying and removing clusters. We 33
refer to them here as Reasenberg’s, Kagan’s, and Zhuang’s methods, respectively. All 34
employ quantitative models of the probability that pairs of earthquakes are causally 35
connected. The probabilities are inferred from statistical study of the distribution of 36
inter-event distances and times as a function of event magnitude. Then combination 37
rules are applied, and all events that can be linked together by a chain of causal 38
connections are considered a ‘cluster’. In Reasenberg’s method, each quake is either 39
unconnected to any cluster, or it belongs to a single numbered cluster. In Kagan’s and 40
Zhuang’s methods, each event is assigned a probability that it was not triggered by 41
another earthquake in the catalog. Reasenberg’s is the simplest, with just two 42
adjustable parameters. In practice most users, including Marzocchi et al. [2003], fix 43
the parameters at “standard” values without further optimization. Kagan’s method has 44
several parameters, but for their southern California catalog Kagan et al. [2006] used 45
parameter values optimized for the Northwest and Southwest Pacific regions [Kagan46
and Jackson, 2000].47
48
Earthquakes are normally labeled as mainshocks, aftershocks, or foreshocks. A 49
mainshock may either be isolated (not part of a cluster) or the largest event within a 50
cluster. It is usually the first event in the cluster, but it may be preceded by one or 51
more foreshocks, smaller by definition than the mainshock. Clearly, the events in a 52
cluster cannot be labeled until all such events shape the cluster. In our terminology, 53
‘triggered events’ are all those after the first in a cluster, and ‘spontaneous 54
3
earthquakes’ include all isolated mainshocks plus the first event in each cluster. Thus, 55
a spontaneous earthquake could be either a mainshock or the first foreshock in a 56
cluster. In the following we will refer to the “spontaneous rate” or just “rate” to mean 57
the model-dependent rate of events after declustering. 58
Seismic studies and earthquake forecasts often assume that spontaneous earthquakes 59
are stationary or time invariant over a long period and estimate the future spontaneous 60
rate directly from the past rate [e.g. Kagan 2007]. However, some have argued that 61
spontaneous earthquakes are not stationary in southern California and have provided 62
different physical explanations [e.g., Press and Allen, 1995; Marzocchi et al., 2003; 63
Selva and Marzocchi, 2005].64
Assessing stationarity is challenging for several reasons. First, the completeness and 65
precision of earthquake catalogs change with time because of seismic network and 66
data analysis changes. Second, declustering is an art, and different approaches can 67
give different results. Third, the definition of stationarity is potentially problematical. 68
A stationary process is one whose statistical properties do not change with time (or 69
space), but the output of this process might show temporal variations. If spontaneous 70
earthquakes are Poissonian, then stationarity implies a constant rate within simply 71
estimated uncertainties. However, Kagan and Jackson [2000], Kagan [2004] and 72
others have shown that declustered earthquakes are better fit by a negative binomial 73
distribution. The negative binomial implies greater variability than a Poisson process, 74
even if both were stationary. Marzocchi et al. [2003] and Selva and Marzocchi [2005] 75
employed non-parametric tests that should not depend on a Poisson or other 76
assumption. However, the robustness of these tests under different conditions may be 77
doubtful: there are many different statistical tests for stationarity whose relative 78
advantages are not well known in the seismological community. Since any statistical 79
test has its own features and there is no single “best test”, it is valuable to see the 80
results from different statistical tests. 81
4
We suppose that spontaneous earthquakes occur as a direct response to tectonic 82
loading, while triggered events are strongly influenced by other earthquakes in the 83
catalog. We take as a null hypothesis that the process causing spontaneous 84
earthquakes is time invariant, or stationary. If an earthquake catalog fails a test for 85
stationarity, we interpret that as suggestive evidence: some external source increased 86
or decreased the earthquake rate. We consider a hypothesis to fail when the 87
significance, or “p value,” is below 0.05. Sir Ronald Fisher thought of the p value as 88
the probability under the null hypothesis of a result more extreme than that actually 89
observed [Rice, 2006]. The p values of all statistical tests in this paper are provided in 90
Tables 2, 3, 4 and 5. Commonly, people test the null hypothesis at 95% confidence 91
level, corresponding to p = 0.05. If a p value is larger than 0.05, we consider that the 92
corresponding null hypothesis cannot be rejected and vice versa. In the following, we 93
will use a shorthand notation to describe many statistical tests of the null hypothesis. 94
“Non-stationary” will describe a declustered catalog for which the null hypothesis 95
failed at the 95% confidence level (significance 0.05� ), and “stationary” will 96
describe a declustered catalog for which the null hypothesis did not fail at that 97
confidence level.98
An hypothesis that apparent rate changes are caused externally gains credibility if 99
results are similar for different : (1)earthquake catalogs, especially those based on 100
different networks, (2) declustering algorithms, (3) magnitude thresholds, (4) subsets 101
of the data, and (5) geographical regions that would be affected by the same external 102
cause.103
In the following sections, we first explain the conclusions drawn by Marzocchi et al. 104
[2003] and Selva and Marzocchi [2005]. Then we test the conclusions in each of these 105
papers separately with different catalogs, magnitude thresholds, declustering methods, 106
and regions of southern and northern California. Finally, we use different earthquake 107
catalogs and Zhuang’s declustering method to show whether spontaneous earthquakes 108
are stationary in all of California. 109
5
110
2.�Apparent�changes�in�southern�California�earthquake�111
behavior�112
Press and Allen [1995] applied both pattern recognition and cluster analysis to 113
earthquakes in southern California and found that the distribution of their focal 114
mechanisms changed around 1970. They inferred that this change might be caused by 115
a small change in the direction of relative motion between the Pacific and North 116
American plates or by stress changes from distant earthquakes. Marzocchi et al. [2003] 117
found that the rate of spontaneous earthquakes in southern California changed 118
significantly in the 1960s. They explained this by the influence of large remote 119
earthquakes like the 1960 Chile and 1964 Alaska events. Selva and Marzocchi [2005] 120
inferred significant changes in both the spontaneous rate and the focal mechanism 121
distribution of spontaneous earthquakes since 1933, and provided a possible 122
explanation by modeling the post-seismic stress perturbation field from the Alaska 123
and Chili earthquakes. 124
Marzocchi et al. [2003] and Selva and Marzocchi [2005] used different catalogs, 125
methods for identifying clusters, statistical tests for rate changes, and slightly different 126
time and space limits. Nevertheless, they agreed with Press and Allen [1995]: the rate 127
of spontaneous earthquakes decreased for a few decades, beginning sometime in the 128
mid- to late 1960s. They further suggested that co-seismic and/or post-seismic 129
Coulomb stress rate changes from the 1960 Chile and 1964 Alaska earthquakes may 130
have caused the apparent earthquake rate change, pointing out that changes in seismic 131
networks and data processing might also play a role. Selva and Marzocchi [2005] also 132
inferred a change in average focal mechanism parameters during the same period, just 133
as Press and Allen [1995] had concluded from a different analysis of different 134
earthquakes. Here we consider only the evidence of earthquake rate changes, not that 135
of focal mechanism variations. 136
6
Earthquake rates depend strongly on magnitude, so completeness is important. 137
Marzocchi et al. [2003] assumed that the southern California catalog is complete at 138
magnitude 4.0 after 1932, based on Hileman et al. [1973]. There are many precedents 139
for that assumption, but a recent study by Felzer [2008] indicates that it is overly 140
optimistic. Felzer [2008] estimated separately the completeness histories for 8 141
different California regions: Northeast (NE), North, San Francisco (SF), Central Coast 142
(CC), Los Angeles (LA), Mojave (MJ), Mid, and the rest of the state. We adopted her 143
results. Figure 1 shows these regions and Table 1 summarizes the estimated 144
completeness in each region as of 1932 (see Tables 5-12 in Felzer [2008] for 145
completeness histories). Felzer found that the magnitude of completeness varies 146
strongly from place to place, and that in 1932 the catalog is complete at magnitude 4.0 147
only for the Los Angeles region.148
149
Examining the rate changes reported by Marzocchi et al. [2003] and Selva and 150
Marzocchi [2005], we asked whether151
1. The conclusions are confirmed by different catalogs? 152
2. The conclusions depend on specific minimum magnitude thresholds? 153
3. The conclusions depend on a specific declustering method? 154
4. The same change occurs in all regions of southern California? 155
5. A similar change occurs in northern California? 156
The first three questions concern the reality of the change itself, while the last two 157
concern the cause. 158
2.1�Testing�Marzocchi�et�al.�[2003]�159
Marzocchi et al. [2003] used the Reasenberg [1985] method with standard parameters 160
to decluster the Southern California Seismic Network (SCSN) catalog [Clinton et al.,161
2006]. They selected earthquakes from 1932 to 2000 of magnitude 4.0 and larger 162
within a rectangular area of southern California shown by the dashed rectangle in our 163
7
Figure 1. They noticed an apparent decrease in the rate of earthquake occurrence, 164
shown by the cumulative plot of isolated earthquakes vs. time in their Figure 9. Using 165
the Wilcoxon test [e.g., Gibbons, 1971], they rejected with 95% confidence the 166
hypothesis that the medians of the annual number of spontaneous events before 1960 167
(1932-1959) and after 1964 (1965-1999) are the same. 168
169
We used the same method to decluster the same catalog in the same region and time 170
interval used by Marzocchi et al. [2003]. The dashed line in Figure 2 shows our result. 171
It resembles their Figure 9 except for two slight differences. First, we found increases 172
in the earthquake rate after declustering, immediately after the 1952 Kern county 173
(magnitude 7.5) and 1992 Landers (magnitude 7.3) earthquakes. They found similar 174
increased rates, but they started slightly before the dates of those earthquakes. We do 175
not know why, but we were working with their published figures, and graphical 176
discrepancies might explain the difference. Second, we found more spontaneous 177
events than they did. However, if we counted only isolated mainshocks (excluding the 178
first event in each cluster), then we agreed almost exactly with their total number. So, 179
in Section 2, only isolated mainshocks are used when we employed Reasenberg’s 180
declustering method. 181
182
We also tested the median annual numbers of isolated events with the Wilcoxon test, 183
as they did. Row 7 of Table 2 shows the parameters and p values of the statistical tests. 184
When we copied their data and methods as closely as possible, we concluded, as they 185
did, that the spontaneous rate apparently decreases significantly after 1965.186
We then carried out some additional calculations to test the robustness of the rate 187
change.188
2.1.1�Test�method�189
In addition to the Wilcoxon test, we used a Kolmogorov-Smirnov (KS) test of the 190
hypothesis that the spontaneous quakes from 1932 to 2000 are stationary in southern 191
8
California. We applied that test to several variants of the declustered catalog, as 192
described below. Results are shown in column 5 of Table 2. When we used the same 193
declustering method and completeness threshold as Marzocchi at al. [2003], (Row 7 194
of Table 2) the hypothesis of stationary occurrence was rejected at the 95% 195
confidence level. This is the same result inferred from the Wilcoxon test. However, 196
for other situations discussed below, the two tests give different results. The 197
Kolmogorov-Smirnov test is a robust statistical test but our results do not depend on it. 198
199
2.1.2�Completeness�Assumption�200
We repeated the calculations using the Reasenberg declustering method, but in 201
calculating the spontaneous rate we counted only earthquakes above the 1932 202
completeness threshold estimated by Felzer [2008]. Results are shown in row 8 of 203
Table 2. In this case the Wilcoxon test does not reject stationarity, while the KS test 204
does. We conclude from this test that the completeness threshold is key to the results. 205
2.1.3�Declustering�Method�206
We used a stochastic declustering method [Zhuang et al., 2002, Zhuang et al., 2004, 207
Zhuang et al., 2005] on the same earthquake data. We will refer to this as Zhuang’s 208
method. It is based on the Epidemic Type Aftershock Sequence, or ETAS, model 209
[Ogata, 1988, 1998; etc.]. In stochastic declustering, one calculates for each event a 210
probability that it was not triggered by previous events. We call that the spontaneous 211
probability. Figure 2 shows the results. Significant differences occur between the 212
dash-dot line (Reasenberg declustering) and solid line (Zhuang declustering) after we 213
consider completeness information in Felzer [2008]. First, the cumulative number of 214
spontaneous events is lower after Zhuang declustering. Second, the spontaneous rate 215
does not increase following the 1952 Kern County and 1992 Landers earthquakes 216
when we used Zhuang’s method, but it did for Reasenberg declustering. The 217
comparison suggests that Reasenberg’s method did not completely remove the 218
9
aftershocks of these two large events. Third, the KS test does not reject stationarity of 219
Zhuang declustering results at the 95% confidence level. It does for the results of 220
Reasenberg declustering. 221
2.1.4�Magnitude�Threshold�222
We repeated the calculations above using the same SCSN catalog but with magnitude 223
threshold 4.7 instead of 4.0. The results are not shown in the tables, because this 224
change had no important effect: after 1964 the rate of spontaneous earthquakes 225
decreased significantly.226
227
However, Jones and Hauksson [1997] drew a different conclusion from that 228
of Marzocchi et al. [2003]. They both used the same catalog, very similar areas, and 229
the same Reasenberg declustering. The primary difference was that Jones and 230
Hauksson [1997] used a lower magnitude threshold of 3.0. Jones and Hauksson [1997] 231
found a high spontaneous rate from January 1945 through July1952, low from August 232
1952 through July 1969, high from August 1969- through August 1992, and low again 233
from September 1992 to through May 1996. The low-rate periods follow the 1952 234
Kern County and 1992 Landers earthquakes. The third period, 1969 - 1992, overlaps 235
substantially with that from 1964 to 1997, in which Marzocchi et al. [2003] found a 236
low rate. Differences in their conclusions might be caused by updates of the SCSN 237
catalog, other processing changes, incompleteness of the catalog at magnitude 3.0, or 238
from subtle effects of the declustering algorithm. It could be that Reasenberg’s 239
algorithm identifies small aftershocks more aggressively than larger ones, leaving 240
periods of apparently low activity following the 1952 and 1992 earthquakes, and thus 241
seemingly higher activity otherwise. There is empirical evidence for the last 242
possibility. From the top curve in Figure 2, we see that at magnitude 4 and larger, the 243
rate of declustered events increases immediately after the earthquakes of 1952 and 244
1992. Jones and Hauksson [1997] found that at magnitude 3 and larger, the rate 245
decreased in those periods. If Reasenberg’s algorithm overcorrected for aftershocks 246
10
then, the remaining periods would appear to have anomalously high rates of 247
spontaneous events. In any case the apparent variations in spontaneous earthquake 248
rate depend on magnitude threshold, as well as the specific declustering method. 249
250
The result of our comparison is that apparent spontaneous rate changes depend only 251
slightly on which test is used. The Kolmogorov-Smirnov and Wilcoxon tests agree in 252
those cases where there is enough data. The spontaneous rate change reported by 253
Marzocchi et al. [2003] is not changed by raising the magnitude threshold from 4.0 to 254
4.7, although Jones and Hauksson [1997] got conflicting results with a threshold of 255
3.0. However, results do depend strongly on the declustering technique. When we 256
declustered using Zhuang’s method, the hypothesis of uniform rate was not rejected 257
by either the Wilcoxon or KS test at the 95% confidence level. 258
2.1.5�Local�Rate�Variations�259
We further examined each of the Felzer completeness zones within southern 260
California. We excised those parts that did not fit in the box used by Marzocchi et al.261
[2003], and considered only regions with at least 20 spontaneous events from 1932 to 262
2000. P values of statistical tests are shown in the lower part of Table 2. We applied 263
both the KS and Wilcoxon tests, in the latter case on the medians of the annual 264
earthquake counts in the periods 1932 - 1959 and 1964 – 1999. When declustered by 265
Reasenberg’s method and when magnitude 4.0 was used as a completeness threshold, 266
the Central Coast and Mojave sub-regions appeared non-stationary by the KS test 267
only. All appeared stationary according to the Wilcoxon test. After Zhuang 268
declustering when Felzer [2008] completeness information was used, only the Mid 269
region appeared non-stationary by the KS test. The Mojave region appeared non-270
stationary according to the Wilcoxon test. No region was unambiguously non-271
stationary, that is, none failed both tests regardless of its completeness criterion or 272
how it was declustered. If the 1960 Chile or 1964 Alaska earthquake caused 273
spontaneous earthquake rate changes, most likely the rate changes would be observed 274
11
in all southern California regions. However, we did not find this no matter which 275
declustering method was used or what completeness information was applied. 276
Furthermore, we found that after declustering with Reasenberg’s method, the 277
spontaneous rate suddenly increased after both 1952 Kern county and 1992 Landers 278
earthquakes in only one region: the Mojave where these large events occurred. We 279
did not find this kind of increase in any one region or throughout southern California 280
after declustering with Zhuang’s method. These facts imply that the sudden increases 281
in 1952 and 1992 are artificial results of incomplete declustering by Reasenberg’s 282
method. 283
2.2�Testing�Selva�and�Marzocchi�[2005]�284
Within a region shown by the dotted polygon in Figure 1, ,Selva and Marzocchi [2005] 285
looked for systematic changes in earthquake rate and focal mechanism, using a recent 286
southern California catalog by Kagan et al. [2006]. The latter used a branching model 287
[Kagan and Jackson., 1991] to compute what they labeled a “mainshock probability” 288
for each event. We refer to this declustering method as Kagan’s method. In fact 289
“mainshock probability” is the probability that each event was not triggered by 290
preceding events in the catalog, so we refer to it here as “spontaneous probability.” 291
Selva and Marzocchi chose a fixed probability, 0.90, above which an earthquake was 292
treated as spontaneous. They performed a change point analysis [Mulargia and Tinti,293
1985] to determine the time such that the rates before and after differ most 294
significantly. They found the estimated change point to be 1959, with a standard 295
deviation of about 5 years. The median rate after 1959 (3.5/year) was lower than that 296
before (5.0/year) at more than 95% confidence. They also performed change point 297
analysis on a function of the earthquake focal mechanisms, finding a change point of 298
about 1969, again with a 5 year standard deviation. They assumed that changes in rate 299
and focal mechanism had the same cause sometime between 1959 and 1969. Selva 300
and Marzocchi offered a hypothesis, supported by numerical calculations, that such 301
changes resulted from a combination of elastic and viscoelastic stress rate changes 302
12
from the 1960 Chile and 1964 Alaska earthquakes. Allowing for uncertainty in the 303
change point and possible time delay for a viscoelastic process to take effect, they 304
concluded that the spontaneous rate was significantly lower after the 1960s than 305
before that decade. 306
307
We checked all the rate calculations of Selva and Marzocchi [2005] except their 308
change point analysis. Some of their descriptions were slightly ambiguous. For 309
example, when they referred to the period 1969 – 2003, did that include both the first 310
and last year and was that range synonymous with “after the 1960s”? For reasonable 311
choices we agreed with their significance tests, but we found that different choices 312
gave different results. For example, the rate for 1969 – 2003 (inclusive) was 313
significantly lower than that for 1933 – 1958, but not when compared to 1933 – 1959.314
315
The data used by Selva and Marzocchi [2005] showed an increased spontaneous rate 316
following the 1952 Kern County and 1992 Landers earthquakes, suggesting that the 317
Kagan and Jackson [1991] method, like Reasenberg’s, does not catch all aftershocks 318
of large earthquakes. This fact does not by itself explain the apparent rate change, but 319
it does cloud the interpretation: local events, as well as any regional effect, are almost 320
certainly affecting the earthquake rate. Selva and Marzocchi [2005] considered 321
temporal changes in detection capability and other non-seismic explanations for the 322
apparent rate change. They dismissed the effect of increased detection capability over 323
time on the grounds that it should increase, rather than decrease, the rate of 324
earthquake occurrence. We believe the effect that better detection will have on 325
‘seeing’ spontaneous events is more complicated. Foreshocks might be missed in the 326
earlier catalog, so that triggering would go undetected and more events would be 327
considered spontaneous. 328
We performed alternative calculations to test the robustness of the inferred rate 329
decrease. First, we calculated earthquake numbers by summing the spontaneous 330
probabilities (“fractional counting”) rather than by counting events over the arbitrary 331
13
0.9 threshold (“binary counting”). For fractional counting, the rate decrease “after the 332
1960s” was not significant at 95%, whether or not 1969 was counted. Second, we 333
declustered the same catalog using Zhuang’s rather than Kagan’s method. A 334
comparison of the cumulative number of spontaneous events vs. time is shown in 335
Figure 3. We performed a KS test of stationarity for the period 1933-2003; the 336
hypothesis was not rejected at the 95% confidence level. Third, we repeated the 337
calculations using only earthquakes above Felzer’s completeness level in each zone. 338
The rate changes appear significant if we copy the methods of Selva and Marzocchi339
[2005], but if we decluster using Zhuang’s method there are no significant rate 340
changes. Again, the most important factor in determining stationarity is the 341
declustering technique.342
Moreover, we applied Zhuang’s method separately to two subsets: 1932-1959 and 343
1970 to 2003. We tested the null hypothesis that spontaneous rate is the same in these 344
two subsets using both Wilcoxon and two-sample KS tests. The p value of the 345
Wilcoxon test is 0.49 and that of the KS test is 0.80, both of which indicate that the 346
hypothesis was not rejected at the 95% confidence level.347
In the passage above, we’ve compared Zhuang’s declustering method with 348
Reasenberg’s for the SCSN catalog (Figure 2) and Zhuang’s method with Kagan’s for 349
the catalog of Kagan et al. [2006](Figure 3). In Figure 4, we compare the results of 350
declustering the southern California catalog of Kagan et al. [2006] with all three 351
methods. The lower magnitude threshold is 4.7, the lower limit of the catalog. We 352
counted cumulative events after 1932, but examined earthquake clusters that started 353
before that date, and assigned probabilities or labels accordingly. For Kagan’s and 354
Zhuang’s methods, we used fractional counting, while for Reasenberg’s method we 355
used traditional binary counting. For Reasenberg’s method, we counted the first event 356
in any cluster as spontaneous. Reasenberg’s method removes the fewest triggered 357
events, leaving the highest number of spontaneous ones. Kagan’s method removes 358
slightly more triggered events. Both left bursts of events, presumably aftershocks, 359
14
following the 1952 Kern County and 1992 Landers earthquakes. By contrast, 360
Zhuang’s method identifies more events as clustered, leaving the fewest spontaneous 361
ones. By eye, Zhuang’s method leaves the most stationary trend, and the statistics 362
bear that out. The spontaneous events identified by Reasenberg’s and Kagan’s 363
methods are non-stationary by the KS test, while those from Zhuang’s method do not 364
fail the KS test at 95% confidence level. By the Wilcoxon test, results are similar 365
except that Kagan’s spontaneous catalog does not fail the stationarity test. 366
3.�Rate�changes�in�northern�California�367
368Northern and southern California share many tectonic features: earthquakes in one 369
might trigger events in the other, and regional tectonic stress changes most likely 370
affect both. For that reason we applied the analysis techniques used above to northern 371
California, all of California, and specific parts of California. We treat those separately 372
because different earthquake catalogs are available and different regions have 373
different completeness histories. 374
375
The SCSN catalog does not cover northern California fully, so we used ANSS there. 376
ANSS is a combines information from the SCSN catalog, the Northern California 377
Seismic Network (NCSN) catalog [http://www.ncedc.org/ncedc/catalog-search.html] 378
and so on. It provides more complete and accurate information in California. 379
Marzocchi et al. [2003] used latitude36.3 N� as the northern boundary of southern 380
California, so we chose it as the southern boundary of northern California. The other 381
boundaries are those of “greater California,” the test region employed in the Regional 382
Earthquake Likelihood Models project [Schorlemmer et al. 2007] and further 383
described in Wang et al. [2009]. “Greater California” extends beyond the political 384
boundaries of California, and it is shown by the solid polygon with larger width in 385
Figure 1. We used both Reasenberg’s and Zhuang’s methods to decluster the ANSS 386
catalog with minimum magnitude 4.0, but in estimating the rate of spontaneous 387
15
earthquakes we counted only those events above the 1932 completeness thresholds 388
estimated by Felzer [2008]. Thus, some earthquakes below the completeness 389
threshold were used to identify triggered events, but none were counted as 390
spontaneous. Figure 5 shows a cumulative count of earthquakes vs. time, and Table 4 391
shows the p values of statistical tests for stationarity. We also tested the stationarity of 392
rates within those parts of the Felzer [2008] completeness zones within northern 393
California. Table 4 shows those results too. For the Reasenberg-declustered catalog, 394
all of northern California and the central coast sub-region fail the KS tests for 395
stationarity. By contrast, all of northern California and all sub-regions are stationary 396
by the Wilcoxon test. However, for the Zhuang-declustered catalog, neither the whole 397
region nor any sub-region is significantly non-stationary by either test. As for 398
southern California, the evidence for variations in rate of spontaneous earthquakes 399
depends on which method is used to recognize clustered events. 400
4.�Rate�changes�in�all�of�California?�401
402After testing the conclusions and explanation of Marzocchi et al. [2003] and Selva403
and Marzocchi [2005], we then asked: can we reject the hypothesis that spontaneous 404
earthquakes throughout California are stationary? To explore this issue, we tried 405
different catalogs and different minimum magnitudes. We used two different catalogs: 406
a unified catalog in California 1800-2007 [Wang et al., 2009] and the ANSS catalog. 407
The advantages of the former are many: it combines 27 different catalogs covering 408
California and thus is more complete; second, it provides more accurate location and 409
magnitude information so important to spontaneous rate estimation; and finally, it 410
includes earthquakes before 1932, data necessary to estimate the spontaneous rate in 411
the 1930s. However, that catalog has a minimum magnitude threshold of 4.7. In order 412
to include smaller earthquakes, ANSS is a very good supplement. We chose 4.0 as the 413
minimum magnitude threshold in the latter. 414
415
16
We used only Zhuang’s declustering because Reasenberg’s and Kagan’s methods 416
apparently do not remove all aftershocks of the 1952 and 1992 earthquakes. We 417
counted only those earthquakes above the magnitude of completeness in each region 418
in northern California, although smaller ones were treated as possible triggers. The 419
solid line in Figure 6 shows the cumulative spontaneous events in California from 420
1932 to 2007 using the unified catalog 1800-2007 [Wang et al., 2009]. By the 421
Kolmogorov-Smirnov test, we could not reject with 95% confidence the hypothesis 422
that spontaneous earthquakes are stationary from 1932 to 2007. We also used the 423
Cramer-von Mises test [Eadie et al., 1971] which gave the same results as the prior 424
test.425
426
The dash-dot line in Figure 6 shows the cumulative spontaneous events in California 427
from 1932 to 2008 in the ANSS catalog. The Kolmogorov-Smirnov test did not reject 428
the stationary hypothesis for 1932 to 2008 at the 95% confidence level. The Cramer-429
von Mises test [Eadie et al., 1971] gave the same results as the Kolmogorov-Smirnov 430
test. Table 5 also shows the KS test results in each region using both catalogs. We can 431
reject the stationary hypothesis only in the Central Coast region, for the ANSS catalog. 432
�433
5��Discussion��434
5.1�Declustering�Methodology��435
�436
Declustering starts with a model for earthquake clustering, a complex physical 437
process. The ambiguity of outcomes documented above stems from the lack of 438
uniqueness in modeling earthquake interactions. Separate clustering models can be 439
evaluated for their internal consistency, their agreement with earthquake data, and 440
their consistency with the subjective views of seismologists. All three declustering 441
methods discussed above have been used frequently, and there have been few reports 442
17
of internal inconsistency. In fact this quality has been little studied. All three methods 443
have parameters optimized using maximum likelihood or related methods for fit to 444
selected data sets, but these parameters might change for different locations and times. 445
All three models have been studied statistically for sensitivity of their parameters to 446
data variations, but standard or default parameters are almost always used in 447
applications to specific earthquake catalogs. The three declustering methods have 448
different numbers of parameters, but so far there has been little research to compare 449
these models against one another. Hainzl et al. [2006] did compare Reasenberg’s 450
method with stochastic declustering: the basis of Zhuang’s method. Using ETAS, they 451
generated synthetic catalogs with a known, constant rate of spontaneous earthquakes. 452
Then they declustered using Reasenberg’s method and found that the distribution of 453
interval times in the declustered catalog was inconsistent with the known input values. 454
The most relevant test will be to measure how models optimized to fit past data will 455
fit future, independent data. In this paper we have applied the third (subjective) 456
criterion in questioning Reasenberg’s and Kagan’s methods because they did not 457
recognize apparent aftershocks of the 1952 and 1992 earthquakes. We don’t have the 458
same complaint about Zhuang’s method, but that is far from proof of its validity. One 459
danger in declustering is circular reasoning. The clustering models make certain 460
assumptions, and it’s risky to conclude that consistency of results with those 461
assumptions validates them. For example, Zhuang’s stochastic declustering method 462
starts with the assumption that the spontaneous rate is constant in time in a first 463
iteration. The assumption is relaxed in later iterations, but the method might be biased 464
towards a constant spontaneous rate. Zhuang [2006] demonstrated by simulation that 465
such a bias is not observed in practice. He constructed simulated catalogs using an 466
ETAS model with a non-stationary background rate, and then declustered using the 467
stochastic method. The spontaneous rate in the output catalog converged to the 468
assumed non-stationary rate. In addition, there are other declustering methods 469
available. For instance, the method of Hainzl et al., [2006] could be used to estimate 470
the rate of spontaneous events and its evolution with time. The technique of Marsan471
and Lengling [2008] provides the probability that each event is a triggered event. It 472
18
would be constructive to compare the results using these declustering methods in 473
future work.474
475
5.2�Stationarity�of�spontaneous�rate�476
Our studies leave the question of stationarity in southern California unresolved. While 477
the declustering methods used by Marzocchi et al. [2003] and Selva and Marzocchi 478
[2005] apparently miss some aftershocks, we could only dismiss them on subjective 479
grounds. We can’t assert that Zhuang’s method correctly separates all spontaneous 480
events from clustered ones. Spurious rate variations could be caused by deficiencies 481
in the declustering models. Other causes might include: changes in detection 482
capability of networks or changes in software or parameters used to determine 483
earthquake magnitudes, inaccurate estimates of catalog completeness, triggering by 484
local earthquakes below the completeness threshold, mistakes in discriminating blasts 485
from earthquakes, etc. We’ve not discussed any of the other causes here, although 486
each deserves attention. The firmest statement we can now make about stationarity is 487
that previous estimates of rate decreases in southern California are not robust.488
5.3�Effects�of�the�1960�Chile�and�1964�Alaska�earthquakes�489
The magnitude 9.5 Chile and 9.2 Alaska earthquakes are among the four or five 490
largest earthquakes since 1900, and it is reasonable to ask if they could affect 491
seismicity anywhere on earth. Given their locations, one would expect the dynamic, 492
static, and viscoelastic stress effects of these earthquakes to be relatively uniform over 493
southern California and indeed over all California. The distances from the epicenter of 494
the Chile earthquake to the centers of southern and northern California are about 495
9,700 km and 10,000 km, respectively. As Cifuentes [1989] estimated, the rupture 496
length of that event was around 920km, so the regions are 10.5 and 10.9 fault 497
dimensions, respectively. From the Alaska epicenter, the distances are 3,700 km to 498
19
southern California and 3,200 km to northern California. As Ichinose et al. [2007] 499
estimated, the rupture length of the Alaska earthquake was around 680km, so the 500
regions are 5.4 and 4.7 fault dimensions, respectively. Thus, the effects of these 501
quakes on southern and northern California seismicity would be at least qualitatively 502
similar. In our study, neither region showed compelling evidence that the Chile and 503
Alaska events reduced spontaneous seismicity. However, spatially consistent behavior 504
should be observed, or inconsistent behavior explained, before concluding that distant 505
events affect regional seismicity. 506
507
6��Conclusions��508
�509
Declustering of earthquake sequences is still a subjective process. We have used 510
Reasenberg’s algorithm, Kagan’s Critical Branching Model, and Zhuang’s stochastic 511
declustering algorithm on several data sets, finding that both the number and apparent 512
stationarity differ considerably among the three methods. Reasenberg’s algorithm, 513
when used on the southern California catalog at magnitude threshold 4.0, leaves in514
concentrations of earthquakes, almost certainly aftershocks, following the 1952 Kern 515
County and 1992 Landers events. The same is true for Kagan’s method for 516
magnitudes 4.7 and larger. That feature does not explicitly explain the longer term 517
variations reported by Marzocchi et al. [2003] and Selva and Marzocchi [2005]. But it 518
implies that those methods do not fully separate spontaneous from triggered 519
earthquakes. Further research on earthquake clustering will be required before 520
spontaneous earthquakes can be confidently isolated, identified, and their properties 521
quantified.522
523
Given conflicting results from different declustering methods, magnitude thresholds, 524
assumptions about completeness, time intervals, and testing methods, the reported 525
decline in spontaneous earthquake rate in southern California after the 1960s is not 526
20
robust. We cannot conclude confidently at this time whether the spontaneous rate 527
decreased in southern California. The suggestion that the rate decrease did occur and 528
was caused by the 1960 Chile and 1964 Alaska events conflicts with fact: the reported 529
rate decrease was not consistent throughout California. 530
531
532
533
534
Acknowledgement535
The Authors are very grateful for Kathleen Jackson’s help on manuscript preparation. 536The authors appreciate support from the National Science Foundation through grant 537EAR-7032928556, as well as from the Southern California Earthquake Center 538(SCEC). SCEC is funded by NSF Cooperative Agreement EAR-0529922 and the U.S. 539Geological Survey (USGS) Cooperative Agreement 07HQAG0008. Comments by 540anonymous reviewers have been helpful in revising the manuscript. Publication 0000, 541SCEC.542
21
References:543
Cifuentes, I. L. (1989), The 1960 Chilean earthquakes, J. Geophys. Res., 94(B1), 665–544
680545
Eadie W. T., D. Drijard, F. E. James, M Roos and B. Sadoulet (1971), Statistical546
Methods in Experimental Physics, North-Holland, Amsterdam. . 547
John F. Clinton, Egill Hauksson and Kalpesh Solanki (2006), an Evaluation of the 548
SCSN Moment Tensor Solutions: Robustness of the Mw Magnitude Scale, Style of 549
Faulting, and Automation of the Method, Bulletin of the Seismological Society of 550
America; 96; 1689-1705; DOI: 10.1785/0120050241 551
Felzer, K. R.(2008), Calculating California seismicity rates, Appendix I in The 552
Uniform California Earthquake Rupture Forecast, version 2 (UCERF 2): U.S. 553
Geological Survey Open-File Report 2007-1437I and California Geological Survey 554
Special Report 203I, 42 pp. 555
556
Gardner, J.K. and L. Knopoff (1974). Is the sequence of earthquakes in southern 557
California, with aftershocks removed, Poissonian? , Bulletin of the Seismological 558
Society of America, 64, 1363-1367 559
560
Gibbons, J. D., Non-parametric Statistical Inference, 306 pp., McGraw-Hill, New 561
York, 1971. 562
563
Hainzl, S., C. Beauval and F. Scherbaum (2006), Estimating background activity 564
based on interevent-time distribution, Bulletin of the Seismological Society of America,565
96, 313-320. 566
567
22
Hileman, J. A., C. R. Allen, and J. M. Nordquist, Seismicity of the southern California 568
region, 1 January 1932 to 31 December 1972, Seismol. Lab., Calif. Inst. of Technol., 569
Pasadena, 1973. 570
571
Ichinose, G., P. Somerville, H. K. Thio, R. Graves, and D. O'Connell (2007), Rupture 572
process of the 1964 Prince William Sound, Alaska, earthquake from the combined 573
inversion of seismic, tsunami, and geodetic data, J. Geophys. Res., 112, B07306, 574
doi:10.1029/2006JB004728.575
576
Jones, L.M. and E. Hauksson (1997), The seismic cycle in southern California: 577
precursor or response?, Geophys. Res. Lett. 24 (4), pp. 469–472.578
579
Kagan, Y.Y. (2004) Short-Term Properties of Earthquake Catalogs and Models of 580
Earthquake Source, Bulletin of the Seismological Society of America; 94; 1207-1228;581
DOI: 10.1785/012003098582
Kagan, Y. Y., D.D. Jackson (1991). Long-term earthquake clustering, Geophysical583
Journal International 104, 117-133 584
585
Kagan, Y, Y, and D. D. Jackson (2000). Probabilistic forecasting of earthquakes, 586
Geophys. J. Int. 143, 438–453.587
588
Kagan, Y. Y., D.D. Jackson, and Y. Rong (2006). A new catalog of southern 589
California earthquakes, 1800-2005, Seismological Research Letters, 77, 30-38 590
591
Kagan, Y. Y., D. D. Jackson, and Y. F. Rong, (2007). A testable five-year forecast of 592
moderate and large earthquakes in southern California based on smoothed seismicity, 593
Seismological Research Letters, 78(1), 94-98. 594
595
23
Marsan, D., and Lengline, I., 2008. Extending earthquakes’ reach through cascading. 596
Science, 319, 1076–1079, doi:10.1126/ science.1148783. 597
598
Marzocchi, W., J. Selva, A. Piersanti, and E. Boschi (2003), on the long term 599
interaction among earthquakes: Some insight from a model simulation, J. Geophys. 600
Res., 108 (B11), 2538, doi:10.1029/2003JB002390. 601
602
Mulargia, F., and S. Tinti (1985), Seismic sample areas defined from incomplete 603
catalogs: An application to the Italian territory, Phys. Earth Planet. Inter., 40, 273– 604
300.605
606
Ogata, Y. (1988): Statistical Models for Earthquake Occurrences and Residual 607
Analysis for Point Processes, Journal of the American Statistical Association, 83, 9–608
27.609
610
Ogata, Y. (1998): Space-time point-process models for earthquake occurrences, 611
Annals of the Institute of Statistical Mathematics, 50, 379–402.612
613
Press, F., and C. Allen (1995), Patterns of seismic release in the southern California 614
region, J. Geophys. Res., 100, 6421– 6430. 615
616Reasenberg, P. (1985), Second-order moment of central California Seismicity, 6171969– 1982, J. Geophys. Res., 90(B7), 5479– 5495.618
619
Rice, A. J. (2006), Mathematical statistics and data analysis, Duxbury Press, India 620
621
Selva, J., and W. Marzocchi (2005), Variations of southern California seismicity: 622
Empirical evidence and possible physical causes, J. Geophys. Res., 110, B11306, 623
doi:10.1029/2004JB003494.624
625
24
Schorlemmer, D., and M. Gerstenberger, S. Wiemer, D. D. Jackson, and D. A. 626
Rhoades, 2007. RELM testing center, in Special Issue on Working Group on Regional 627
Earthquake Likelihood Models (RELM), Seismol. Res. Lett., 78(1), 30. 628
629
Utsu, T. (1969), Aftershock and earthquake statistics. I: Some parameters which 630
characterize an aftershock sequence and their interrelations, J. Fac. Sci., Hokkaido 631
Univ., Ser. VII, Geophys., 3, 129–195. 632
633
Wang, Q, D.D. Jackson, and Y. Y. Kagan (2009). California earthquakes, 1800-2007: 634
a unified catalog with moment magnitudes, uncertainties and focal mechanisms. 635
Seismological Research Letters, 80, 446-457.636
637
Zhuang, J., Y. Ogata and D. Vere-Jones (2002). Stochastic declustering of space-time 638
earthquake occurrences, Journal of the American Statistical Association, 97, 369-380 639
640
Zhuang, J., Y. Ogata, and D. Vere-Jones (2004), Analyzing earthquake clustering 641
features by using stochastic reconstruction, J. Geophys. Res., 109, B05301, 642
doi:10.1029/2003JB002879643
644
Zhuang, J., C.-P. Chang, Y. Ogata, and Y.-I. Chen (2005), A study on the 645
spontaneous and clustering seismicity in the Taiwan region by using point process 646
models, J. Geophys. Res., 110, B05S18, doi:10.1029/2004JB003157. 647
648
Zhuang J. (2006) Second-order residual analysis of spatiotemporal point processes 649
and applications in model evaluation. Journal of the Royal Statistical Society: Series 650
B (Statistical Methodology), 68, 635-653. doi: 10.1111/j.1467-9868.2006.00559.x. 651
25
652Index Name Complete magnitude
1 Northeast region(NE) 5.7
2 North region (North) 5.6
3 San Francisco region (SF) 4.5
4 Central Coast region (CC) 4.1
5 Mid region (Mid) 4.2
6 Los Angeles region (LA) 3.9
7 Mojave region (MJ) 4.1
8 The rest of the state (RS) 6.0
Table 1: Regions and magnitude completeness as of 1932 based on the estimation of 653
Felzer [2008] 654
26
655Region Southern California
Polygon Dash rectangular in Figure 1
Catalog SCEDC catalog
Minimum magnitude 4.0
Time period 1932-1999
Polygon Decluster
method
Completeness N KS Test
p value
N1 N2 W Test
p value
Reasenberg 4.0 1092 3.5E-14 572 438 0.001
Reasenberg Felzer 464 4.5E-4 208 227 0.32
Whole
Region
Zhuang Felzer 296 0.49 136 142 0.090
Reasenberg 4.0 87 0.016 30 54 0.86
Reasenberg Felzer 34 0.22 17 15 0.57
Central
Coast
Zhuang Felzer 27 0.90 14 12 0.54
Reasenberg 4.0 41 0.45 19 21 0.93
Reasenberg Felzer 23 0.0026 6 17 0.18
Mid
Zhuang Felzer 13 0.011 2 11 0.12
Reasenberg 4.0 130 0.28 41 73 0.73
Reasenberg Felzer 130 0.28 41 73 0.73
LA
Zhuang Felzer 94 0.81 30 49 0.86
Reasenberg 4.0 366 4.0E-4 176 163 0.076
Reasenberg Felzer 276 9.2E-5 134 122 0.049
Mojave
Zhuang Felzer 154 0.24 75 69 0.047
Table 2: Tests of stationarity in southern California using methods of Marzocchi et al.,656
[2003] (Row 7) and our results using alternate methods: “Polygon”: “Central Coast”, 657
“Mid”, “LA”, and “Mojave” indicate regions suggested by Felzer [2008]; 658
Completeness: “Felzer” indicates that threshold in Table 1 is used. “N” means total 659
number of spontaneous earthquakes used; “KS Test p value” is the p value of 660
Kolmogorov-Smirnov test of hypothesis that spontaneous earthquakes are stationary 661
from 1932 through 1999. “N1” and “N2” are numbers of spontaneous earthquakes 662
from 1932 through 1959; and 1965 through 1999. Bolt number indicates the null 663
hypothesis is rejected at 95% confidence level. “W test p value” is the p value of 664
27
Wilcoxon test of hypothesis that the median of the annual number of spontaneous 665
earthquakes is the same in both periods. Bolt number indicates the null hypothesis is 666
rejected at 95% confidence level. Note that only isolated events are considered as 667
spontaneous events when Reasenberg declustering method is used. 668
28
669Region Southern California
Polygon Dotted polygon in Figure 1
Catalog RELM catalog in southern California
Minimum magnitude 4.7
Time period 1933-2003
Polygon Declustering
method
Completeness Counting
method
N KS Test
p value
N1 N2 W Test
p value
Kagan 4.7 Binary 275 0.015 121 130 0.047
Kagan 4.7 Fraction 305 0.003 134 145 0.093
Kagan Felzer Fraction 231 9.5E-6 99 115 0.18
Whole
Region
Zhuang Felzer Fraction 113 0.61 44 55 0.43
Kagan 4.7 Binary 45 0.0012 16 27 0.85
Kagan 4.7 Fraction 52 2.8E-4 19 31 0.80
Kagan Felzer Fraction 52 2.8E-4 19 31 0.80
LA
Zhuang Felzer Fraction 29 0.43 11 16 0.97
Kagan 4.7 Binary 139 1.3E-5 58 69 0.12
Kagan 4.7 Fraction 159 7.9E-7 69 78 0.077
Kagan Felzer Fraction 159 7.9E-7 69 78 0.077
Mojave
ETAS Felzer Fraction 75 0.41 29 36 0.47
Table 3: Tests of stationarity in southern California using method of Selva and 670
Marzocchi [2003] (Row 7) and our results using alternate methods: Polygon: “LA” 671
and “Mojave” indicate regions suggested by Felzer [2008]; Completeness: “Felzer” 672
indicates that threshold in Table 1 is used. Counting method: “Binary” means that 673
event is counted as 1 if spontaneous probability exceeds 0.9 and as 0 otherwise; 674
“Fraction” means that earthquakes are counted by summing spontaneous probabilities 675
29
of all events. “KS Test p value” is the p value of Kolmogorov-Smirnov test of 676
hypothesis that spontaneous earthquakes are stationary from 1933 through 2003. “N1” 677
and “N2” are numbers of spontaneous earthquakes from 1933 through 1958; and 1969 678
through 2003. Bolt number indicates the null hypothesis is rejected at 95% confidence 679
level. “W test p value” is the p value of Wilcoxon test of hypothesis that the median 680
of the annual number of spontaneous earthquakes is the same in both periods. Bolt 681
number indicates the null hypothesis is rejected at 95% confidence level. For other 682
notation see caption to Table 2. 683
30
684Region Northern California
Polygon In the north of 36.3 N� in “greater California”
Catalog ANSS catalog
Minimum magnitude 4.0
Time period 1932-1999
Polygon Declustering
Method
Completeness N KS Test
p value
N1 N2 W Test
p value
Reasenberg Felzer 235 2.0E-5 80 142 0.74
Reasenberg+ Felzer 289 8.1E-9 92 184 0.38
Whole
Region
Zhuang Felzer 159 0.43 75 75 0.078
Reasenberg Felzer 57 0.88 26 28 0.60
Reasenberg+ Felzer 69 0.94 28 37 0.98
San
Francisco
Zhuang Felzer 54 0.99 25 27 0.53
Reasenberg Felzer 94 4.6E-8 21 71 0.51
Reasenberg+ Felzer 111 6.6E-12 21 88 0.44
Central
Coast
Zhuang Felzer 40 0.72 17 23 0.60
Reasenberg Felzer 68 0.12 22 40 0.30
Reasenberg+ Felzer 83 0.08 28 49 0.30
Mid
Zhuang Felzer 44 0.46 21 20 0.49
Table 4: Tests of stationarity in northern California using various methods. The 685
chosen time intervals and notation are the same as for Table 2. Reasenberg+ means 686
both isolated events and first events in clusters are considered as spontaneous events 687
when the Reasenberg declustering method is used. 688
31
689
Region California California
Polygon “greater California” “greater California”
Catalog Wang et al. [2009] ANSS
Minimum magnitude 4.7 4.0
Time period 1932-2007 1932-2007
Total number of events 551 1775
N KS test
p value
N KS test
p value
Whole Region 240 0.61 528 0.11
San Francisco 46 0.62 58 0.63
Central Coast 36 0.95 105 0.025
Mid 26 0.97 59 0.25
Los Angeles 32 0.99 109 0.87
Mojave 70 0.49 175 0.37
Table 5: Test of the hypothesis that spontaneous earthquakes are stationary in 690
California: KS test is Kolmogorov-Smirnov test of hypothesis that the spontaneous 691
earthquakes are stationary from 1932 to 2007 at 95% confidence: Other notation is the 692
same as that of Table 3. 693
694
32
695Figure 1: Polygons used in this paper: The largest solid line polygon encloses “greater 696
California”; the dash line rectangular shows the region used by Marzocchi et al.697
[2003]; the dotted line polygon shows the region used by Selva and Marzocchi [2005]. 698
699
33
700Figure 2: Cumulative spontaneous earthquakes at or above magnitude 4.0 in southern 701
California. Dashed line shows our results using the same catalog and declustering 702
method of Marzocchi et al. [2003].That curve matches closely that in Figure 9 of 703
Marzocchi et al. [2003]. The dash-dot line shows our results using the same catalog 704
and declustering, but counting only events at or above the relevant Felzer [2008] 705
completeness threshold. The solid line shows our results using Zhuang’s stochastic 706
declustering on the same catalog, also with magnitude threshold 4.0 and using Felzer 707
zone completeness. Note that only isolated events are considered as spontaneous 708
events when Reasenberg declustering method is used. 709
34
710Figure 3: Cumulative sum of spontaneous events at or above magnitude 4.7 in 711
southern California: The dashed line (top curve) shows the fractional count 712
(cumulative sum of spontaneous probabilities) using the catalog of Kagan et al.713
[2007]. The double dashed line shows the results of the same data using integer 714
counting of events with probability 0.9 and greater, as in Selva and Marzocchi [2005]. 715
The dash-dot line is the same as the top curve, except that earthquakes not satisfying 716
Felzer [2008] completeness conditions are not counted. The solid line shows the 717
fractional count of the same data, except declustered by the method of Zhuang et al.718
[2005] and omitting events below the Felzer completeness threshold. 719
35
720Figure 4: Cumulative spontaneous earthquakes in Southern California from 1932 to 721
2000 using Kagan et al. [2006] earthquake catalog with minimum magnitude 4.7. The 722
solid line shows the result of Zhuang’s declustering, the dash line shows the result of 723
Kagan’s declustering and the dash-dot line shows the result of Reasenberg’s 724
declustering. The selection region is latitude 32 to 37, longitude -122 to -114, which is 725
used by Kagan et al. [2006]. Note that both isolated events and first events in clusters 726
are considered as spontaneous events when the Reasenberg declustering method is 727
used.728
729
36
730Figure 5: Cumulative spontaneous earthquakes at or above magnitude 4.0 in northern 731
California from the ANSS catalog. The double dashed line shows our results using 732
Reasenberg [1985] and both isolated events and first events in clusters are considered 733
as spontaneous events. The dash-dot line shows our results using Reasenberg [1985] 734
declustering and only isolated events are considered as spontaneous events. The solid 735
line shows our results using Zhuang declustering on the same catalog. Earthquakes 736
below Felzer completeness threshold are omitted in both cases. 737
37
738
Figure 6: Cumulative spontaneous earthquakes in California from 1932 to 2007 using 739
Zhuang declustering: The solid line shows the results using the ANSS earthquake 740
catalog with minimum magnitude 4.0. The dash-dot line shows the results using the 741
earthquake catalog from Wang et al. [2009] with minimum magnitude 4.7. 742
Earthquakes below Felzer completeness threshold are omitted in both cases. 743744745
