Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS...
Transcript of Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS...
![Page 1: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/1.jpg)
Comparison of ETAS parameter estimates across1
different global tectonic zones2
Chu, A.1, Schoenberg, F.P.2, Bird, P.3, Jackson, D.D.3, and Kagan, Y.Y.33
4
1 The Institute of Statistical Mathematics5
6
2 Department of Statistics, University of California, Los Angeles7
8
3 Department of Earth and Space Sciences, University of California, Los Angeles9
10
Email: [email protected]
Postal address: 10-3 Midori-cho,12
Tachikawa, Tokyo 190-8562, Japan13
1
![Page 2: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/2.jpg)
Abstract14
Branching point process models such as the ETAS (Epidemic-Type Aftershock Se-15
quence) models introduced by Ogata (1988, 1998) are often used in the description,16
characterization, simulation, and declustering of modern earthquake catalogs. The17
present work investigates how the parameters in these models vary across different tec-18
tonic zones. After considering divisions of the surface of the Earth into several zones19
based on the plate boundary model of Bird (2003), ETAS models are fit to the occur-20
rence times and locations of shallow earthquakes within each zone. Computationally,21
the EM-type algorithm of Veen and Schoenberg (2008) is employed for the purpose of22
model fitting. The fits and variations in parameter estimates for distinct zones are com-23
pared. Seismological explanations for the differences between the parameter estimates24
for the various zones are considered, and implications for seismic hazard estimation25
and earthquake forecasting are discussed.26
Keywords: branching processes, earthquakes, epidemic-type aftershock sequence model,27
space-time point process models, maximum likelihood, global earthquake prediction, plate28
tectonics.29
30
1 Introduction31
Researchers have recently defined global tectonic zones according to geophysical character-32
istics such as crust types and relative plate velocities using plate boundary models (Bird,33
2003; Bird and Kagan, 2004; Kagan et al., 2010). The primary purpose of this study is34
2
![Page 3: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/3.jpg)
to investigate the different properties and patterns of seismicity in these different zones by35
exploring the fit of space-time branching point process models to the data within each zone.36
Similarities and differences between the background rates and clustering behavior of earth-37
quake occurrences in different tectonic zones may shed light on the interplay between tectonic38
forces and observed seismicity rates.39
Similar investigations were performed by Kagan et al. (2010), using the branching point40
process model developed by Kagan (1991). Here, we explore the fit of Epidemic-Type Af-41
tershock Sequence (ETAS) models of Ogata (1998) within each tectonic zone. Such models,42
which have become widely used in the study of the properties of earthquake clustering and43
in the forecasting of seismicity, are based on the notion that earthquakes occur at some44
background rate, those background events trigger aftershocks, and those aftershocks trigger45
subsequent aftershocks, etc. ETAS models have been used extensively in the analysis of local46
catalogs (e.g. Ogata, 1998; Schoenberg, 2003; Zhuang et al., 2005; Ogata and Zhuang, 2006;47
Helmstetter et al., 2007), and of global catalogs (Lombardi and Marzocchi, 2007; Marzocchi48
and Lombardi, 2008), but we are unaware of their prior use in the analysis of global tectonic49
zones such as those delineated by Bird (2003).50
This article is organized as follows. We first present a brief description of the delineation51
of tectonic zones and global earthquake data in Section 2. Section 3 describes the ETAS52
model of Ogata (1998), as well as model diagnostics and the computational implementation53
used. Section 4 discusses the results of estimating ETAS models within each tectonic zone,54
including the maximum likelihood estimates (MLEs) of the parameters in the model, their55
standard errors, fitted triggering functions, estimates of the Gutenberg-Richter constant and56
branching ratio. Section 5 presents statistical diagnostics based on the estimated conditional57
3
![Page 4: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/4.jpg)
intensity, and diagnostics using comparisons between the empirical aftershock rates and fitted58
triggering functions. Concluding remarks and discussion of future work follow in Section 6.59
2 Tectonic zones and earthquake catalog60
2.1 Tectonic zones61
Tectonic zones are related to the 7 plate boundary classes defined in global plate model62
PB2002 (Bird, 2003), but zones are defined differently so as to allow for earthquake clas-63
sification even if location, depth, and focal mechanism are poorly known. As displayed in64
Figures 1, there are four major zones (Kagan et al., 2010):65
1. Active continent (including continental parts of all orogens of PB2002, plus continental66
plate boundaries of PB2002).67
2. Slow-spreading ridges (oceanic crust, spreading rate < 40 mm/yr, including trans-68
forms).69
3. Fast-spreading ridges (oceanic crust, spreading rate ≥ 40 mm/yr, including trans-70
forms).71
4. Trenches (including incipient subduction, and earthquakes in outer rise or upper plate).72
In what follows, we refer to the zone consisting of the plate interiors, i.e. the rest of the73
Earth’s surface occupied by the non-colored areas in Figure 1, as zone 0.74
It is reasonable to anticipate different branching behaviors between the zones, and our75
present definition of zones is designed with consideration for practical matters that will76
4
![Page 5: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/5.jpg)
permit such testing to be conducted relatively easily. Zones are defined here as collections77
of polygons delineating the surface areas into which epicenters of shallow earthquakes may78
fall. The zones are not divided based on the depths of the earthquakes because these depths79
are usually poorly estimated unless there is local station control, so we follow Bird et al.80
(2002) in limiting our analysis simply to shallow earthquakes of depth less than 70 km. The81
resulting partition produces five major zones, and each zone contains a sufficient number of82
earthquakes within the time period considered (1973-2006) to enable branching models to83
be accurately and reliably estimated.84
Our tectonic zones are defined by objective rules implemented in a program because this85
is reproducible, easy to explain, easy to revise, and not very subject to procedural error.86
Our program assigns tectonic zones to grid points rather than attempting to draw boundary87
curves about each zone. It loops through latitudes (in 0.1o steps) and also loops through88
longitudes (in 0.1o steps) to create a grid of zone index integers which identify the zone at89
each grid point. The decision algorithm for each point is detailed in the Appendix of Kagan90
et al. (2010). The necessary data are available in digital form: elevation from ETOPO591
(Anonymous, 1988), age of seafloor from Mueller et al. (1997), plate boundaries and Euler92
poles from Bird (2003).93
The output is a relatively compact (6 to 12 MB) representation, which is shown in Figure94
1. The gridded zone integers may be downloaded from http://bemlar.ism.ac.jp/wiki/95
index.php/Bird’s_Zones.96
Note that some tectonic zones are the unions of non-continuous patches and may include97
quite different types of earthquakes. For instance, strike-slip and normal-faulting earth-98
quakes are contained in the same mid-ocean spreading ridges and continental rift zones,99
5
![Page 6: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/6.jpg)
and similarly, along trenches, subduction-related earthquakes are analyzed collectively along100
with back-arc-spreading earthquakes.101
2.2 Earthquake catalog102
The data used for our analysis were retrieved from the PDE (Preliminary Determination103
of Epicenters) database (USGS, 2008). The time window we study is January 1, 1973 to104
December 31, 2006. Only shallow earthquakes (0 to 70 kilometers in depth) are considered.105
The shallow events constitute approximately 80% of all events in the catalog. Summaries of106
catalog size and area are shown in Table 1. For each zone’s analysis, a lower magnitude cutoff107
threshold of mt 5.0 is used, following the example of Kagan et al. (2010). The PDE dataset108
with lower magnitude threshold 5.0 consists of mostly (97%) body wave (55%), surface wave109
(8%), moment magnitude (31%), and small percentage of local magnitude (3%).110
Table 1 displays each zone’s surface area, in proportion to that of the globe, and the pro-111
portion of all observed events occurring in each zone. Among these zones, zone 4 (trenches)112
has the largest number of events (64.5% out of total 40504 events) while only occupying113
2.98% of the global surface area. Zone 1 (active continent) also contains a disproportion-114
ate number of events, with 16.6% of the events in the global catalog and only 6.58% of the115
global surface area. Zones 1 and 4 are unions of irregularly shaped polygons, with one or two116
larger polygons corresponding to the orogens, and several small and strip-shaped polygons117
corresponding to the plate boundaries. Zones 2 (slow-spreading ridges) and 3 (fast-spreading118
ridges) are similar to one another in number and shape. They each have about 7% of the119
total events (7.47% for zone 2 and 6.35% for zone 3), and occupy a small portion of the120
6
![Page 7: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/7.jpg)
global surface area (3.09% for zone 2 and 1.65% for zone 3). Zones 2 and 3 have very long,121
narrow shapes completely constituted by plate boundaries. Lastly, zone 0 (plate-interior),122
the regime excluding zones 1, 2, 3 and 4, occupies 43.8% of the whole global surface and123
contains 5.06% of the global events.124
3 Model and methods125
3.1 Model126
The ETAS model is a type of Hawkes point process model; such models are also sometimes127
called branching or self-exciting point processes. For a temporal Hawkes process, the con-128
ditional rate of events at time t, given information Ht on all events prior to time t, has the129
form130
λ(t|Ht) = µ+∑i:ti<t
g(t− ti), (1)
where µ > 0 is the background rate, g(u) ≥ 0 is the triggering function dictating the rate131
of aftershock activity associated with a prior event, and∫∞0 g(u)du < 1 in order to ensure132
stationarity (Hawkes, 1971).133
These models were called epidemic by Ogata (1988) because of their natural branching134
structure: an earthquake can produce aftershocks, and these aftershocks produce their own135
aftershocks, etc. An example is the time-magnitude ETAS model of Ogata (1988), which136
has magnitude-dependent triggering function137
g(ui;mi) =K0
(ui + c)pea(mi−mt), (2)
7
![Page 8: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/8.jpg)
where ui = t − ti is the time elapsed since earthquake i, K0 > 0 is a normalizing constant138
governing the expected number of direct aftershocks triggered by earthquake i, mi denotes139
the recorded magnitude of earthquake i, and mt is the lower cutoff magnitude for the earth-140
quake catalog. The term K0/(ui + c)p describing the temporal distribution of aftershocks is141
known as the modified Omori-Utsu law (Utsu et al., 1995).142
Ogata (1998) extended the ETAS model to describe the space-time-magnitude distribu-143
tion of earthquake occurrences, introducing circular or elliptical spatial functions into the144
triggering function so that the squared distance between an aftershock and its triggering145
event follows a Pareto distribution. For instance, a form for the conditional rate proposed146
in Ogata (1998) is147
λ(t, x, y|Ht) = µ+∑i:ti<t
g(t− ti, x− xi, y − yi,mi),148
where149
g(t− ti, x− xi, y − yi,mi) = K0(t− ti + c)−p((x− xi)2 + (y − yi)2
ea(mi−mt)+ d)
−q
, (3)
and where (xi, yi) represent the Cartesian coordinates of earthquake i.150
With the triggering function (3), the spatial distribution of aftershocks interacts dramat-151
ically with mainshock magnitude. This interaction is sometimes referred to as magnitude152
scaling, since a typical feature of the model and of earthquake catalogs is a gradual widening153
of the spatial-temporal aftershock distribution as the magnitude of the mainshock increases154
(Utsu et al., 1995).155
Typically, in local catalogs, one calculates the squared epicentral distance in the Cartesian156
plane, via D2i = (x − xi)2 + (y − yi)2 as in (3), but for global seismicity it is important to157
8
![Page 9: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/9.jpg)
account for the sphericity of the Earth. Hence in this analysis we replace this term in (3)158
with the squared great circle distance between (x, y) and (xi, yi).159
Ogata (1998) and others have extended the model (1) to include a spatially inhomoge-160
neous background rate µ(x, y). Note that in our analysis, we fit a homogeneous background161
rate in each zone, since our primary goal is to examine the differences in clustering behavior162
and overall rates between the various zones, rather than the spatial heterogeneity within163
each zone. Further, ETAS models are fit to the data within each zone individually; no trig-164
gering between zones is assumed. Implications and extensions to more complex models are165
discussed in Section 6.166
3.2 Methods167
The parameters in the ETAS model may be estimated by maximizing the log-likelihood:168
logL(θ) =∑i
logλ(ti, xi, yi|Hti)−∫ T
0
∫ y1
y0
∫ x1
x0
λ(t, xi, yi)dxdydt (4)
where θ = (µ, a, K0, c, p, d, q) is the vector of parameters to be estimated and [x0, x1] ×169
[y0, y1] × [0, T] is the space-time window where points in the marked point process dataset170
(xi, yi, ti, mi) are observed (Ogata, 1998; Daley and Vere-Jones, 2003, ch. 7). Maximum171
likelihood estimates of the parameters in the ETAS model are obtained based on the EM172
(Expectation-Maximization type algorithm) proposed in Veen and Schoenberg (2008), with173
modification to accommodate the magnitude scaling in (1) and the use of great circle distance174
mentioned in Section 3.1. The algorithm of Veen and Schoenberg (2008) is similar to the175
original, EM algorithm of Dempster, Laird and Rubin (1977), which is now widely used to176
9
![Page 10: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/10.jpg)
obtain estimates of parameters in the presence of unobserved random variables influencing177
the likelihood. The EM algorithm is based on an expectation (E) step where the expectation178
of the likelihood given the unobserved variables is computed and a maximization (M) step179
where the parameters maximizing the expected likelihood computed in the E step are found.180
The above steps are typically iterated until a certain tolerance is reached. In the EM-type181
estimation procedure proposed by Veen and Schoenberg (2008), the branching structure182
dictating which earthquake triggered which aftershocks comprises the unobserved random183
variables and is estimated according to the ETAS model in each E step. This assignment184
changes in each iteration as the parameters in the ETAS model change following each M185
step. The idea is similar to the stochastic reconstruction idea of Zhuang, Ogata and Vera-186
Jones (2002, 2004), in which the ETAS model is similarly used to assign a random branching187
structure to an observed earthquake catalog.188
Note that the ETAS model allows assignments where a triggered event has larger magni-189
tude than its parent event and that despite the fact that the branching structure estimated190
in each iteration of the EM algorithm will often be quite erroneous, the resulting estimation191
procedure nevertheless produces quite stable and accurate parameter estimates, as shown by192
Veen and Schoenberg (2008).193
For zones 1, 2, 3, and 0, the time window 1973 to 2001 was used for computation of194
MLEs (maximum likelihood estimates). Due to concerns associated with computer memory195
limitations, the time window 1999 to 2006 was used to obtain the MLEs of zone 4, the196
trench zone; we have ensured that in all of these cases, the MLEs appear already to have197
converged and stabilized within these shortened time windows. The standard error for each198
parameter estimate may be estimated using the square root of the diagonal elements of the199
10
![Page 11: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/11.jpg)
inverse of the Hessian of the log-likelihood (Ogata, 1978). As noted in Wang et al. (2010)200
and Kagan et al. (2010), such asymptotic standard error estimates may be unreliable in201
the presence of strong correlations between parameter estimates, making the statistical sig-202
nificance of differences between parameter estimates for different tectonic zones difficult to203
assess. One method for obtaining more accurate standard errors would be to simulate the204
ETAS model repeatedly, after fitting its parameters by MLE (maximum likelihood estima-205
tion), and subsequently re-estimating the model’s parameters for each simulation. One may206
then use the standard deviation of these estimates as an approximation of the standard error.207
Such repeated simulation and re-estimation of the models is extraordinarily computationally208
intensive, however, and approximate standard errors for each parameter estimate obtained209
individually by investigating the square root of the inverse Hessian when optimizing each210
parameter individually are used as an approximation.211
In addition to studying the behavior of ETAS parameter estimates in the different tectonic212
zones, one may also investigate other properties of the models, such as their information213
gain relative to the stationary Poisson model. The information gain is defined as (Kagan214
and Knopoff, 1977; Kagan, 1991):215
I =`− `0N
(5)216
where ` is the logarithm of the likelihood for the ETAS model, `0 is the logarithm of the217
likelihood for the stationary Poisson process, and N is the catalog size.218
Since the ETAS model contains the Poisson process as a special case when the triggering219
function is 0, the information gain must necessarily be non-negative, and since the difference220
11
![Page 12: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/12.jpg)
between ETAS and the stationary Poisson model is that the former is capable of modeling221
intense clustering behavior, higher values of the information score in this case generally222
signify increased clustering in the earthquake catalog.223
Another comparison of the triggering properties in different zones can be made by in-224
vestigating the branching ratio. Letting m = mi −mt, then G, the total expected number225
of directly triggered aftershocks per earthquake of magnitude mi within each zone, can be226
defined as the integral of (3) over time and space:227
G(m) =∫ ∞0
∫ −∞−∞
∫ −∞−∞
g(t, x, y|Ht) dxdydt = πK0(p− 1)−1c1−p(q − 1)−1d1−qeam. (6)228
In the subcritical case, the expected value E(G) =∫G(m)f(m)dm coincides with the229
branching ratio (Veen, 2006), and estimates of this branching ratio for each zone are discussed230
in Section 4.2.231
In addition to the aforementioned assessment methods, we compare the estimated condi-232
tional intensity at a grid of locations, integrated over time, with the corresponding observed233
rate of earthquake occurrences within a small rectangle around each location in the grid.234
Letting (0, T ) denote the observed time window,235
∫ T
0λ(t, x, y)dt = µT +K0
∑i
∫ T−ti
0(t+ c)−p(Ri
2/ea(mi−mt) + d)−qdt (7)
= µT +K0
∑i
(Ri2/ea(mi−mt) + d)−q(cp−1 − (T − ti + c)p−1)/(p− 1) (8)
where Ri is the distance between the location (x, y) and a previous earthquake’s location,236
(xi, yi); particularly, the great circle distance between (x, y) and (xi, yi) is used in the global237
12
![Page 13: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/13.jpg)
case. Comparison of the integrals in (8) with observed seismicity rates for various locations238
(x, y) may be assessed directly, or via the various standardizations of these differences, as239
discussed in Baddeley et al. (2005). As a further diagnostic, we compare the estimated240
triggering function g in (3), with the empirical, observed aftershock rate. The procedure241
to obtain the empirical aftershock rate is the following: we select earthquakes in a given242
fixed magnitude range and consider their productivity of direct aftershocks by inspecting243
subsequent earthquakes within some small time-space window within the catalog. Note that244
since the background rate is much smaller than the triggering of the conditional intensity in245
(1) at such locations and times, the empirical triggering function and g should be similar in246
such regions if the model is appropriate.247
4 Results248
4.1 Maximum likelihood estimates of ETAS parameters249
The MLEs (maximum likelihood estimates), their standard errors, and the log-likelihoods250
of the ETAS model and the reference model (stationary Poisson) are shown in Table 2.251
The information gains are all larger than 1, showing apparent improvement in fit due to the252
ETAS model compared to the stationary Poisson model, and indicating significant clustering253
in each zone. Zone 0 (plate-interior) has the largest information gain, 4.20, due to the254
more pronounced spatial and temporal heterogeneity of seismicity within this zone. The255
information gains in zones 1 (active continent) and 4 (trenches) are also somewhat large256
(2.62 for zone 1 and 2.24 for zone 4), due to the high level of clustering in these two zones.257
13
![Page 14: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/14.jpg)
Zones 2 (slow-spreading ridges) and 3 (fast-spreading ridges) have the smallest information258
gains (1.79 for zone 2 and 1.20 for zone 3), as the seismicity within these zones is relatively259
homogeneous and less clustered than the other zones, so the gain due to the ETAS model,260
relative to a stationary Poisson model, is comparatively small. Note that the analyses in261
Kagan et al. (2010) use the same tectonic zone partitions and calculate the information262
gain relative to an inhomogeneous Poisson model. Kagan et al. (2010) also use different263
parametrization of their critical branching model. Here we apply homogeneous models and264
compute the information gain relative to a homogeneous Poisson model.265
The estimated background rate, µ, differs substantially from one zone to another. Not266
surprisingly, the estimate is largest in zone 4, which has by far the highest seismicity rate,267
as seen in Table 1. Zone 3 has the second largest estimated rate of background seismicity,268
but the estimate of this rate in zone 3 is only one fourth of the corresponding estimated rate269
in zone 4. The estimated background rates in zones 1 and 2 are somewhat similar to one270
another. The fact that the estimated background seismicity rate in zone 4 is approximately271
500 times that in zone 0 demonstrates the tremendous variability in overall seismicity from272
one tectonic zone to another, with the vast majority of background events concentrating in273
the trench zone.274
The MLEs of other parameters can similarly be used to summarize patterns of seismicity,275
including levels of clustering, swarms, and spatial triggering behavior, within the different276
tectonic zones. The exponential term in (2) is known as the productivity law in seismology,277
in which the parameter a governs the relationship between the magnitude of an earthquake278
and its expected number of generated offspring and is also useful in characterizing earth-279
quake sequences quantitatively in relation to their classification into seismic types (Mogi,280
14
![Page 15: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/15.jpg)
1963; Utsu, 1970; Ogata, 1988). Smaller estimates of a in zones 2 and 3 are characteristic281
of earthquake swarms, so the relatively low estimates of a in zones 2 and 3 appear to indi-282
cate a correspondingly high incidence of swarming in these tectonic zones. Conversely, the283
estimates of a are largest in zones 1 and 4, indicating a large disparity in aftershock produc-284
tivity between smaller and larger events. Lastly, a is moderately sized in zone 0, indicating285
relatively moderate distinction between large and small events in terms of the productivity286
of subsequent aftershocks in this zone.287
The parameters c and p govern the temporal decay of aftershock activity. The modified288
Omori formula describes the frequency of aftershocks per unit time interval since a preceding289
event as n(t) = K0/(t + c)p (Utsu, 1961). As seen in Table 2, the values of p are similar290
across the zones; this stability indicates the seismic activity in the different tectonic zones is291
somewhat similar in terms of the rate at which the aftershock behavior after an earthquake292
decreases over time. Note that while p seems similar across the zones, the estimate of c in zone293
0 is somewhat higher than that in the other zones. Larger values of c indicate a more gradual294
decaying rate of seismicity following an event, according to the modified Omori formula. The295
estimates of c are much smaller in Zones 2, 3, and 4 than the other zones, indicating that296
these ridges, oceanic transform zones and trenches appear to have sharp temporal decays297
in aftershock activity following earthquakes. Conversely, the apparent aftershock activity298
decays much more slowly and gradually in active continent zones and especially in interplate299
regions, where little aftershock activity is observed even in the close temporal proximity of300
an event.301
The parameter q governs the spatial distribution of aftershocks. In general, smaller q302
corresponds to a more gradual spatial decay of aftershock productivity (long range decay)303
15
![Page 16: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/16.jpg)
while larger q indicates that the triggering effect is mainly confined to a relatively small304
area around an earthquake (short range decay). Our results indicate that, among the major305
zones, zones 1 and 4 have relatively far-reaching triggering effects, and conversely, zones306
2 and 3 have triggering effects that reach a smaller area. Additionally, the value of q for307
zone 0 is much smaller than the corresponding estimates in the major zones, indicating that308
aftershock activity is much more gradually spatially decaying in interplate regions, according309
to the fitted ETAS model. This is likely due in large part to faulting being attributed to310
clustering in the ETAS model, as explained further below.311
Along with q, the square root of d indicates the scale, in km, of the spatial decay in the312
generation of aftershocks. That is, if at a certain time after a particular event of magnitude313
mi, the rate of aftershocks at distance R = 0 from the event is some value s, then at distance314
√dea(mi−mt)/2 the rate decreases to 2−qs, i.e. the rate decreases by a factor of 2−q. As seen315
from Table 2, among the major zones, the estimate of d is smallest in the active continental316
earthquake zone and highest in the ridges and oceanic transform zones. Note that, for two317
models with similar values of q, a higher value of d indicates more gradual spatial decay in318
clustering as one moves further from the mainshock. However, such interpretations can be319
misleading without simultaneously regarding the corresponding estimates of q; zone 3, for320
instance, has the largest estimate of d, indicating more gradual spatial decay with distance,321
but also the largest estimate of q, which indicates sharper spatial decay. The cumulative322
effects of these two parameters on the spatial decay function are explored below.323
K0 is a normalizing constant governing the total expected number of aftershocks per324
earthquake (Ogata 1988, 1998). Estimates of K0 vary dramatically between the various325
tectonic zones, with the highest observed triggering rates in zones 2 and 3. This is quite326
16
![Page 17: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/17.jpg)
counterintuitive, since one would expect higher productivity of aftershocks in the more active327
zones. Additionally, the estimate K0 in zone 0 is approximately the same as that in zone328
1 and zone 4. As noted in Veen (2006), the variance of K0 may be very large due to the329
flatness of (1) and thus large changes in K0 can correspond to very small improvements in330
the log-likelihood (4). In addition, the temporal and spatial components of the triggering331
function g are densities and integrate to unity when integrated over infinite time and space,332
but when integrated over relatively small, thin zones, such as zones 2 and 3 in particular, the333
integral of the spatial component will often deviate substantially from one. This complicates334
the interpretation of K0, since the expected number of aftershocks observed within the given335
zone will deviate markedly from K0 in such cases. Hence it may be preferable to interpret the336
model’s triggering density in total rather than interpret each parameter estimate individually.337
This is explained further below.338
Figure 2 shows both the relationship between the estimated triggering function g and339
∆R or (∆R)2, where ∆R denotes the distance between a mainshock and its aftershock. The340
triggering component is much more pronounced in the active zone (zone 1), as expected,341
compared to the other zones, but this triggering decays vary rapidly as the distance from342
the mainshock increases. Zone 0’s estimated triggering function is noticeably larger than343
that of the other zones. A plausible explanation for the strangely high degree of triggering344
in zone 0 lies in the extreme inhomogeneity of the Earth’s seismicity. Recall that the fitted345
ETAS model has a homogeneous background rate, µ. Within zones 1-4, the seismicity is346
relatively homogeneous, but in zone 0, the ETAS model with homogeneous background rate347
tends to assign a low background rate in order to accommodate areas in zone 0 with no348
seismicity at all, and as a result, the clusters of seismicity that one observes along faults gets349
17
![Page 18: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/18.jpg)
assigned to the triggering portion of the model, although many of the earthquakes along350
these faults might more reasonably be thought of as background events.351
We now return to our concern of the relationship between parameter estimates for a given352
fitted model in a given tectonic zone. Since c and p are both time-related, and d and q are353
both space-related, one might expect strong correlations for these pairs of parameters. As354
noted from simulation studies in Schoenberg et al. (2010), such estimates can be strongly355
correlated. In Figure 3, we observe significant correlations among the pairs (d, q), (d, K0),356
(K0, q), and (K0, information gain), while significant correlation is not observed for some357
other pairs (e.g. for a, Figure 4).358
4.2 Gutenberg-Richter constant, branching ratio359
Maximum likelihood estimates of the Gutenberg-Richter constant β and its corresponding360
b-value differ between tectonic zones, as shown in Table 3. The probability density function361
of m = mi − mt, derived from the Gutenberg-Richter law is (Gutenberg and Richter 1944):362
f(m) =βe−βm
1− e−β(MmaxGR −mt)
(9)
where MmaxGR is defined as the pre-specified maximum magnitude, and the exponent β relates363
to the b-value as β = bln(10) ≈ 2.3b (Veen, 2006). Larger values of β in the above truncated364
exponential density of m indicate more rapid decay in the density function as magnitude365
increases, implying a larger proportion of smaller events. This is the case in zone 2. Zone 3,366
by contrast, has a smaller estimate of β, indicating a smaller proportion of small events.367
Note that a < β for all the tectonic zones, and the difference β − a is largest in zones 2,368
18
![Page 19: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/19.jpg)
3 and 0, implying that the smallest earthquakes contribute most to the overall seismic rate369
(Helmstetter 2003, Sornette and Werner 2005a, 2005b), especially in zones 2, 3 and 0. This370
agrees with the observation in Section 4.1 that zones 2 and 3 tend to contain more swarms371
rather than clusters of clearly distinct mainshocks and aftershocks.372
Table 3 displays the estimated branching ratio (expected number of triggered events per373
mainshock) for the fitted models. Zone 0 has the largest estimated number of aftershocks374
per mainshock. Zone 3 appears to have the fewest aftershocks per mainshock, in agreement375
with the results in Kagan et al. (2010). Among the major zones of primary interest, zone 4376
has the highest estimated branching ratio. This is not surprising because zone 4 is the most377
seismically active regime.378
4.3 Triggering of aftershocks379
Figure 5 shows how the estimated branching ratio, G, based on equation (6), varies as a380
function of mainshock magnitude for different tectonic zones. The curves for zones 2 and381
3 are very similar; the curves corresponding to zones 1 and 4 are also similar. Zone 0382
exhibits a phenomenon between these two types. For zones 1 and 4, a typical M6 event383
would have about 2 expected aftershocks of magnitude ≥ 5.0; a typical M7 event would have384
about 3 expected aftershocks of this size; an M8 event would have about 6 or 7 expected385
aftershocks with M ≥ 5.0; and an M9 event has about 18 or 19 expected aftershocks with386
magnitude ≥ 5.0 within these zones. The number of expected aftershocks increases much387
more dramatically with mainshock magnitude in zones 1 and 4, compared to zones 2, 3388
and 0. For zones 2 and 3, one would expect a typical M6 or M7 event to have only about389
19
![Page 20: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/20.jpg)
one aftershock greater than magnitude 5.0; a M8-9 event would have about two expected390
aftershocks of magnitude ≥ 5.0 in this zone. There is no dramatic change in the number391
of expected aftershocks as the magnitude increases. For zone 0, a typical M6 or M7 event392
would have about 1 or 2 expected aftershocks of magnitude ≥ 5.0, which is somewhat similar393
to zones 2 and 3; a typical M8 or M9 event would have about 3 or 4 direct aftershocks of394
this size. In summary, the aftershock rates of zones 2, 3 and 0 are much more gradual than395
the other zones. This observation coincides with the implication mentioned in Sections 4.1396
and 4.2 that among the major zones, zones 2 and 3 have primarily swarm-type seismicities,397
and zones 1 and 4 have relatively clear distinctions between mainshocks and aftershocks.398
The spatial and temporal ranges of a typical event’s direct (first-generation) aftershocks are399
shown in Figure 5. The top right panel in Figure 5 shows the time (in years) to cover a400
certain proportion, ρ, of direct aftershocks. All the curves are quite similar for ρ < 75%; in401
all zones, at least 75% of the expected direct aftershocks would be expected to occur within 1402
year of the corresponding triggering earthquake. The zones differ, however, for larger values403
of ρ. For example, we expect earthquakes in zone 4 to have 85% of their direct aftershocks404
within 1 or 2 years; an event in zone 1 would have 85% of its expected aftershocks within 5405
years; for zones 0 and 2, 85% of the direct aftershocks are expected to occur within 10 years406
of the mainshock; and for zone 3, one expects it to take nearly 10 years for merely 80% of407
the direct aftershocks to occur. Among the major zones, zones 2 and 3 have less strongly408
clustered seismicity compared to the other zones.409
Figure 5 also shows the spatial range expected to cover a certain proportion of direct410
aftershocks for a typical M6 event. The curves of zones 1, 2, 3 and 4 show that up to 90%411
of the direct aftershocks are within 100 km of their triggering earthquakes, while almost all412
20
![Page 21: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/21.jpg)
aftershocks are within 200 km of their corresponding mainshocks. Zone 0 behaves similarly413
to the other zones except at the upper tail of the percentage of aftershocks to be contained.414
To contain more than 95% of the direct aftershocks, the radius around a mainshock would415
have to be larger than 200 km. Among the major zones, the aftershock radii in zone 2 are the416
smallest, and zone 4 has the largest spatial radii. This is an indication that zone 4 exhibits417
the strongest triggering, so that mainshocks in zone 4 can trigger events to somewhat further418
distances than earthquakes in the other zones. Note that the radii for zone 0 are even larger419
than those for zone 4, which may largely be attributable to the large area of zone 0.420
The bottom right panel in Figure 5 shows the radius (in km) required to contain 95% of421
the expected direct aftershocks for a typical event. The curves of zones 2 and 3 are similar;422
an M7, M8 or M9 event would have 95% of its expected aftershocks within approximately423
100 km. For zones 1 and 4, an M7 event would have 95% of its expected aftershocks within424
150 km; an M8 event would have 95% of its expected aftershocks within 150 to 200 km; and425
in the case of zone 4, we see somewhat larger radii. For zone 1, an M9 event would have426
95% of its expected aftershocks within 300 to 350 km; and an M9 event in zone 4 would427
have 95% of its expected aftershocks within 400 km. Zone 0 is somewhat distinct from the428
other zones for events of M5, M6 and M7 (95% radii are approximately 100 km, 150 km and429
200 km, respectively), but similar to zones 1 and 4 for events of M8 and M9 (95% radii are430
approximately 250 km and 300 km, respectively). Our estimates of the sizes of aftershock431
zones are considerably wider than those estimated in previous research (Abercrombie 1995,432
Kagan et al. 2010). Below (Section 6) we try to explain this ETAS result and its difference433
from other estimates of a focal zone size.434
21
![Page 22: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/22.jpg)
5 Model diagnostics435
5.1 Estimated conditional intensity in space436
The fitted models in each zone may most directly be inspected by examining spatial plots437
of the integrated conditional intensity. In Figure 6, we present an example for each zone:438
1. a sub-region of zone 1 (active continent): Southern California, (−122, −114) in longi-439
tude by (32, 37) in latitude.440
2. a sub-region of zone 2 (slow-spreading ridges): oceanic ridge in the North Atlantic441
Ocean, (−47, −42) in longitude by (20, 30) in latitude.442
3. a sub-region of zone 3 (fast-spreading ridges): oceanic ridge to the south of Australia,443
(145, 150) in longitude by (−60 , −55) in latitude.444
4. a sub-region of zone 4 (trenches): east of Honshu, Japan, (140 , 145) in longitude by445
(30, 40) in latitude.446
The approximation 1o ≈ 111 km, also used in Bird (2003), is employed here.447
One sees in Figure 6 that areas of high estimated intensity correspond to areas of high448
seismicity, and the spatial distribution of the triggering functions appear to fit the data well.449
5.2 Triggering function versus empirical aftershock rate450
Another basic diagnostic tool is to compare the triggering function to the empirical aftershock451
rate. Figure 7 shows the spatial and temporal distribution of all events of magnitude at least452
5.0 occurring within a given space-time window around any event with magnitude in a given453
22
![Page 23: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/23.jpg)
range. For example, for zone 1, we consider in Figure 7 all events occurring after an event454
of estimated magnitude approximately 6.0.455
Note that the estimated background rates for all the zones depicted are small compared456
with the contributions from the triggering functions (see Table 2), so most of the events457
depicted in Figure 7 are attributed by the ETAS model to aftershocks. One can see the458
steep temporal decrease in seismicity following an earthquake, in agreement with the Omori459
law. Spatially, one sees that zone 1 and zone 2 have most of their aftershocks within a 50460
to 100 km radius of the mainshock, but in zone 4, radii of up to 200 km are required to461
contain most of the aftershocks. The spatial plots in Figure 7 do not indicate any strong462
departures from isotropy, at least not in the aggregate over all earthquakes within a zone,463
although in zone 1, aftershocks in the North-South direction appear to be somewhat more464
prevalent. Note that locally, the spatial patterns of aftershocks may be highly anisotropic,465
but that in the aggregate over an entire global zone, when the aftershocks are plotted relative466
to North-South and West-East directions rather than relative to local fault directions, they467
appear nearly isotropic in Figure 7.468
The empirical densities shown in Figure 7 may be compared to the expected densities469
according to the fitted ETAS model, and this comparison is shown in Figure 8. We observe470
that the empirical and expected aftershock density curves coincide very well in each of the471
zones.472
23
![Page 24: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/24.jpg)
6 Conclusions473
We have found that the ETAS models appear to fit rather well to each of the major tectonic474
zones, despite the fact that the estimates of parameters in the ETAS models differ across the475
zones. Most of these differences in parameter estimates can be attributed to the differences476
in overall seismic rates for different zones. Zones 1 (active continent) and 4 (trenches) for477
instance, have the highest overall rates of seismicity and also appear to have the most intense478
estimated triggering functions. Zones 2 (slow-spreading ridges) and 3 (fast-spreading ridges),479
by contrast, appear to have lower background rates, have estimated triggering functions that480
decay more sharply spatially, and generally have parameter estimates consistent with swarms481
rather than traditional mainshock-aftershock clustering.482
As pointed out in Schoenberg et al. (2010), maximum likelihood estimates of ETAS483
parameters may be significantly biased due to the lower cutoff mt. Although the same484
cutoff magnitude of 5.0 is applied to each zone, the bias resulting from this choice of cutoff485
may affect different zones differently due to the varying sparsity of the catalog within each486
zone. A similar sort of truncation bias may be introduced by the irregularly shaped spatial487
windows delineating the different tectonic zones. This may have contributed to biases in the488
parameter estimates, especially for the spatial parameters d and q. Further study is needed489
in order to determine the biases introduced by spatial windows for parameters in ETAS490
models. The time windows selected here, by contrast, appear not to be a major source of491
bias in our parameter estimates; we have tested different time windows of various lengths492
and obtained very similar results for all of the parameters, including the temporal aftershock493
decay parameters c and p. However, the correlations between parameter estimates appear494
24
![Page 25: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/25.jpg)
to be significant, and these correlations can also bias parameter estimates and estimates of495
standard errors.496
Another major source of error in the present analysis is due to the use of homogeneous497
background rates. We have elected to use a homogeneous background rate model because498
of our focus on the triggering behavior in the different tectonic zones. Furthermore, vari-499
ous different methods for estimating inhomogeneous background rates have been proposed,500
including kernel methods and Bayesian smoothing methods, and each may provide substan-501
tially different estimates; we desired our conclusions not to depend critically on our decision502
to use a particular choice of background rate estimation technique. However, our results es-503
pecially for zone 0 appear to depend critically on our choice of homogeneous background rate,504
and a similar study using inhomogeneous ETAS models may be an important direction for505
future work. In addition, one may consider larger models allowing triggering between zones.506
The study of such models with interactions is an important concern for future research. As507
noted in Bird (2003), very small plates within the orogens are likely to be identified in the508
near future, so further modeling of seismicity within single orogens may yield increasingly509
accurate estimates and forecasts of seismicity in the future.510
25
![Page 26: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/26.jpg)
Data and Resources511
All data used in this paper came from published sources listed in the references.512
26
![Page 27: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/27.jpg)
References513
Abercrombie, R.E. (1995). Earthquake source scaling relationships from −1 to 5 ML using514
seismograms recorded at 2.5-km depth. J. Geophys. Res., 100, 24, 015-24, 036.515
Anonymous, Digital Relief of the Earth, Data Announcement [CD-ROM], 88-MGG-02, Natl.516
Geophys. Data Cent., Boulder, Colo., 1988.517
Baddeley, A., R. Turner, J. Møller, and M. Hazelton (2005). Residual analysis for spatial518
point processes. J. R. Statist. Soc. B, 67, Part 5, pp. 617-666.519
Bird, P., Y.Y. Kagan, and D.D. Jackson (2002). Plate tectonics and earthquake potential520
of spreading ridges and oceanic transform faults, in: Plate Boundary Zones, AGU521
Geodynamics Series Monograph Volume 20, eds. S. Stein and J. T. Freymueller, 203-522
218 (doi: 10/1029/030GD12).523
Bird, P. (2003). An updated digital model of plate boundaries, Geochemistry Geophysics524
Geosystems, 4(3), 1027, doi:10.1029/2001GC000252.525
Bird, P., and Y.Y. Kagan (2004). Plate-tectonic analysis of shallow seismicity: apparent526
boundary width, beta, corner magnitude, coupled lithosphere thickness, and coupling527
in seven tectonic settings, Bull. Seismol. Soc. Amer., 94(6), 2380-2399.528
Daley, D., and D. Vere-Jones (2003). An Introduction to the Theory of Point Processes, 2nd529
ed., Springer: New York.530
Dempster, A., N. Laird, and D. Rubin (1977). Maximum likelihood from incomplete data531
via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1), 1-38.532
27
![Page 28: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/28.jpg)
Gutenberg, B., and C.-F. Richter (1944). Frequency of earthquake in California, Bull. Seis-533
mol. Soc. Amer., 34, 185-188.534
Hawkes, A.G. (1971). Point spectra of some mutually exciting point processes. Journal of535
the Royal Statistical Society, Series B, 33(3), 438-443.536
Helmstetter, A. (2003). Is earthquake triggering driven by small earthquakes?, Phys Rev.537
Lett., 91, 10.1103/PhysiRevLett. 01.058501.538
Helmstetter, A., Y.Y. Kagan, and D.D. Jackson (2007). High-resolution time-independent539
forecast for M > 5 earthquakes in California, Seismological Research Letters, 78, 59-67540
Kagan, Y.Y. (1991). Likelihood analysis of earthquake catalogues, Geophys. J. Int., 106(1),541
135-148.542
Kagan, Y.Y., P. Bird, and D.D. Jackson (2010). Earthquake patterns in diverse tectonic543
zones of the globe, accepted by Pure Appl. Geoph., (The Frank Evison Volume),544
167(6/7), in press, doi: 10.1007/s00024-010-0075-3.545
Kagan, Y., and L. Knopoff (1977). Earthquake risk prediction as a stochastic process, Phys.546
Earth Planet. Inter., 14, 97-108.547
Lombardi, A.M., and W. Marzocchi (2007). Evidence of clustering and nonstationarity in548
the time distribution of large worldwide earthquakes. J. Geophys. Res., 112, B02303.549
doi:10.1029/2006JB004568.550
Marzocchi, W., and A.M. Lombardi (2008). A double branching model for earthquake551
occurrence. J. Geophys. Res., 113, B08317. doi:10.1029/2007JB005472.552
28
![Page 29: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/29.jpg)
Mogi, K. (1963). Some discussions on aftershocks, foreshocks and earthquake swarms: the553
fracture of a semi-finite body caused by an inner stress origin and its relation to the554
earthquake phenomena. Bull. Earthquake Res. Inst., 41, pp. 615-658.555
Mueller, D., W. R. Roest, J.-Y. Royer, L. M. Gahagan, and J. G. Sclater (1997). Digital556
isochrons of the world’s ocean floor, J. Geophys. Res., 102(B2)., 3211-3214.557
Ogata, Y. (1978). The asymptotic behavior of maximum likelihood estimators for stationary558
point processes. Annals of the Institute of Statistical Mathematics, 30, 243-261.559
Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for560
point processes. Journal of the American Statistical Association, 83, 9-27.561
Ogata, Y. (1998). Space-time point process models for earthquake occurrences. Annals of562
the Institute of Statistical Mathematics, 50, 379–402.563
Ogata Y., and J. Zhuang (2006). Space–time ETAS models and an improved extension.564
Tectonophysics, 413(1-2), 13-23.565
Schoenberg, F.P. (2003). Multi-dimensional residual analysis of point process models for566
earthquake occurrences. JASA, 98(464), 789-795.567
Schoenberg, F.P., A. Chu, and A. Veen (2010). On the relationship between lower magnitude568
thresholds and bias in ETAS parameter estimates. J. Geophys. Res., 115, B04309,569
doi:10.1029/2009JB006387.570
Sornette, D., and M.J. Werner (2005a). Apparent clustering and apparent background571
earthquakes biased by undetected seismicity. J. Geophys. Res., 110(B9), B09303.572
29
![Page 30: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/30.jpg)
Sornette, D., and M.J. Werner (2005b). Constraints on the size of the smallest triggering573
earthquake from the ETAS Model, Baath’s Law, and observed aftershock sequences.574
J. Geophys. Res., 110(B8), B08304.575
U.S. Geological Survey. Preliminary determination of epicenters (PDE) (2008). US Dep. Of576
Inter., Natl. Earthquake Inf. Cent., http://neic.usgs.gov/neis/epic/epic global.html.577
Utsu, T. (1961). A statistical study on the occurrence of aftershocks, Geoph. Magazine, 30,578
521-605.579
Utsu, T. (1970). Aftershocks and earthquake statistics (II) - Further investigation of af-580
tershocks and other earthquake sequences based on a new classification of earthquake581
sequences. J. Fac. Sci. Hokkaido Univ., Ser.VII, 3, 197-266.582
Utsu, T., Y. Ogata, and R.S. Matsu’ura (1995). The centenary of the Omori formula for a583
decay law of aftershock activity. Journal of Physics of the Earth, 43, 1-33.584
Veen, A. (2006). Some methods of assessing and estimating point processes models for585
earthquake occurrences. PhD thesis, University of California, Los Angeles.586
Veen, A., and F. Schoenberg (2008). Estimation of space-time branching process models in587
seismology using an EM-type algorithm. Journal of the American Statistical Associa-588
tion, 103(482), 614-624.589
Wang, Q., F.P. Schoenberg, and D.D. Jackson (2010). Standard Errors of Parameter Esti-590
mates in the ETAS Model, ms, accepted by Bull. Seismol. Soc. Amer.591
Zhuang, J., Y. Ogata, and D. Vere-Jones (2002). Stochastic declustering of space-time592
30
![Page 31: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/31.jpg)
earthquake occurrences. Journal of the American Statistical Association, 97(458),593
369-380.594
Zhuang, J., Y. Ogata, and D. Vere-Jones (2004). Analyzing earthquake clustering features595
by using stochastic reconstruction. J. Geophys. Res., 109, B05301.596
Zhuang J., Y. Ogata, and D. Vere-Jones (2005). Diagnostic analysis of space-time branching597
processes for earthquakes. Chap. 15 (Pages 275-290) of Case Studies in Spatial Point598
Process Models, Edited by Baddeley A., Gregori P., Mateu J., Stoica R. and Stoyan599
D., Springer-Verlag, New York. 320 pages.600
31
![Page 32: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/32.jpg)
Tables601
32
![Page 33: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/33.jpg)
Zone Area Proportion No. events No. events
(x 10 7 km2) of area in zone in zone / total
1. Active continent 3.36 6.58% 6728 16.6%
2. Slow-spreading ridges 1.58 3.09% 3028 7.47%
3. Fast-spreading ridges 0.844 1.65% 2574 6.35%
4. Trenches 1.52 2.98% 26125 64.5%
0. Plate-interior 43.8 85.8% 2049 5.06%
Total 51.0 100% 40504 100%
Table 1: Areas and PDE catalog sizes for each tectonic zone. Time window is January 1, 1973
to December 31, 2006. Only shallow events (≤ 70 km in depth) with reported magnitudes
of at least 5.0 are considered.
602
33
![Page 34: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/34.jpg)
Estimates Zone 1 Zone 2 Zone 3 Zone 4 Zone 0
Act-Con. Slow Fast Trench Inter.
a 0.980 0.180 0.324 0.942 0.439
(1/magnitude) (0.0179) (0.0543) (0.0530) (0.0181) (0.0466)
c 0.0781 0.0319 0.0307 0.0408 0.497
(days) (0.00341) (0.00282) (0.00320) (0.00195) (0.0396)
d 69.3 227 327 141 69.0
(km2) (0.958) (4.35) (6.91) (2.06) (2.60)
K0 0.539 23.3 77.6 1.33 0.169
(events/day/km2) (0.00963) (0.664) (2.78) (0.0233) (0.00469)
p 1.19 1.16 1.15 1.25 1.21
(0.00309) (0.00414) (0.00488) (0.00430) (0.00440)
q 1.91 2.33 2.49 1.95 1.54
(0.00322) (0.00462) (0.00542) (0.00296) (0.00449)
µ x 10−9 7.63 8.43 15.6 61.7 0.131
(events/day/km2) (0.132) (0.239) (0.398) (0.775) (0.00542)
Log-lik.(ETAS) -109803 -52071 -44562 -380005 -37876
Log-lik.(Poisson) -127451 -57488 -47674 -438746 -46508
Info. gain 2.62 1.79 1.20 2.24 4.20
Table 2: Maximum likelihood estimates of ETAS parameters. Standard errors (diagonal
terms in the Hessian matrix) are reported in parentheses. ’Log-lik (ETAS)’ and ’Log-like
(Poisson)’ refer to the log-likelihood for the ETAS and homogeneous Poisson models, respec-
tively.
34
![Page 35: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/35.jpg)
Zone a β b Branching ratio
1. Active continent 0.980 2.73 1.18 0.521
2. Slow-spreading ridges 0.180 3.15 1.37 0.481
3. Fast-spreading ridges 0.324 2.42 1.05 0.386
4. Trenches 0.942 2.69 1.17 0.559
0. Plate-interior 0.439 3.07 1.33 0.690
Table 3: Estimates of parameters a and β governing the Gutenberg-Richter distribution of
magnitudes, as well as the corresponding b-value (b = β/ln(10)) and expected branching
ratio.
35
![Page 36: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/36.jpg)
Figure Captions603
36
![Page 37: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/37.jpg)
Figure 1: Tectonic zones plotted with world map (Mercator projection). Blue: zone 1 (active
continent), green: zone 2 (slow-spreading ridges), yellow: zone 3 (fast-spreading ridges), red:
zone 4 (trenches), white: zone 0 (plate-interior).
Figure 2: Estimated triggering function g (triggered events / day / km2), fit by maximum
likelihood, as a function of (a) spatial radius or of (b) squared radius. Zone 1: active
continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,
zone 0: plate-interior.
Figure 3: Scatterplots of estimated ETAS parameters as a function of estimates of K0,
for each zone. The numbers in the plots indicate the corresponding zone numbers. The
estimates for zones 0 and 1 nearly overlap in the plot of the estimate of d versus the estimate
of K0. Zone 1: active continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges,
zone 4: trenches, zone 0: plate-interior.
37
![Page 38: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/38.jpg)
Figure 4: Scatterplots of estimated ETAS parameters as a function of estimates of a, for each
zone. The numbers in the plots indicate the corresponding zone numbers. Zone 1: active
continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,
zone 0: plate-interior.
Figure 5: Summaries of the estimated triggering functions for each zone. (a) The expected
number of direct (1st generation) aftershocks of an earthquake of a given magnitude, as a
function of that magnitude, according to the fitted ETAS model for each zone. (b) The
expected time required to contain a given proportion of direct aftershocks, according to the
fitted ETAS model for each zone. (c) The expected spatial range (in km) required to contain
a given proportion of direct aftershocks of a typical M6 event, according to the fitted ETAS
models. (d) The radius (in km) of a range expected to contain 95% of the direct aftershocks
of a typical earthquake of a given magnitude, as a function of that magnitude, according
to the fitted ETAS model for each zone. Zone 1: active continent, zone 2: slow-spreading
ridges, zone 3: fast-spreading ridges, zone 4: trenches, zone 0: plate-interior.
Figure 6: Estimated conditional intensities according to ETAS (in color, on logarithmic
scale) and recorded earthquakes (black dots) for particular regions. The events are shallow
events (≤ 70 km deep) of magnitude ≥ 5.0. (a) A sub-region of zone 1 (active continent) in
California with 122 events in 1973-2006. (b) A sub-region of zone 2 (slow-spreading ridges),
an oceanic ridge in the North Atlantic Ocean, with 113 events in 1973-2006. (c) A sub-region
of zone 3 (fast-spreading ridges), an oceanic ridge South of Australia, containing 86 events
in 1973-2006. (d) A sub-region of zone 4 (trenches), East of Honshu, Japan, containing 185
events in 1999-2006.
38
![Page 39: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/39.jpg)
Figure 7: Histograms (left) and scatterplots (right) of events within 30 days and occurring
within the given great circle distance (GCD) of a given event of recorded magnitude. (a)
and (b): Magnitude 6.0 for zone 1, the active continent. (c) and (d): Magnitude 5.3 for zone
2, the slow-spreading ridge zone. (e) and (f): Magnitude 6.0 for zone 4, the trench zone.
Figure 8: Estimated triggering density g (solid curves) and empirical rate (dotted curves) of
events occurring within 30 days and within a given spatial radius, following an earthquake
of a given magnitude. (a) mainshocks of M6.0 in zone 1 (active continent). (b) mainshocks
of M5.3 in zone 2 (slow-spreading ridges). (c) mainshocks of M5.3 in zone 3 (fast-spreading
ridges). (d) mainshocks of M6.0 in zone 4 (trenches). (e) Mainshocks of M5.3 in zone 0
(plate-interior).
39
![Page 40: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/40.jpg)
Figures604
40
![Page 41: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/41.jpg)
Fig
ure
1:T
ecto
nic
zones
plo
tted
wit
hw
orld
map
(Mer
cato
rpro
ject
ion).
Blu
e:zo
ne
1(a
ctiv
eco
nti
nen
t),
gree
n:
zone
2
(slo
w-s
pre
adin
gri
dge
s),
yellow
:zo
ne
3(f
ast-
spre
adin
gri
dge
s),
red:
zone
4(t
rench
es),
whit
e:zo
ne
0(p
late
-inte
rior
).
41
![Page 42: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/42.jpg)
(a) .
5 10 15 200.0e
+00
5.0e
−06
1.0e
−05
1.5e
−05
2.0e
−05
radius(km)
g
Zone 1Zone 2Zone 3Zone 4Zone 0
(b) .
0 100 200 300 4000.0e
+00
5.0e
−06
1.0e
−05
1.5e
−05
2.0e
−05
radius^2(km^2)
g
Zone 1Zone 2Zone 3Zone 4Zone 0
Figure 2: Estimated triggering function g (triggered events / day / km2), fit by maximum
likelihood, as a function of (a) spatial radius or of (b) squared radius. Zone 1: active
continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,
zone 0: plate-interior.
42
![Page 43: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/43.jpg)
0 20 40 60 80
0.2
0.4
0.6
0.8
1.0
K0
a
0
1
2
3
4
0 20 40 60 80
0.1
0.3
0.5
K0
c
0
12 34
0 20 40 60 80
100
200
300
K0
d
01
2
3
4
0 20 40 60 80
1.16
1.20
1.24
K0
p
0
1
23
4
0 20 40 60 80
1.6
2.0
2.4
K0
q
0
1
2
3
4
0 20 40 60 80
0e+0
03e
−08
6e−0
8
K0
mu
01 2
3
4
0 20 40 60 80
1.5
2.5
3.5
K0
info
rmat
ion
gain
0
1
2
3
4
Figure 3: Scatterplots of estimated ETAS parameters as a function of estimates of K0, for
each zone. The numbers in the plots indicate the corresponding zone numbers. The estimates
for zones 0 and 1 nearly overlap in the plot of the estimate of d versus the estimate of K0.
Zone 1: active continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone
4: trenches, zone 0: plate-interior.
43
![Page 44: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/44.jpg)
0.2 0.4 0.6 0.8 1.0
0.1
0.3
0.5
a
c
0
12 3 4
0.2 0.4 0.6 0.8 1.0
100
200
300
ad
0 1
2
3
4
0.2 0.4 0.6 0.8 1.0
020
4060
80
a
K0
0 1
2
3
4
0.2 0.4 0.6 0.8 1.0
1.16
1.20
1.24
a
p
0
1
23
4
0.2 0.4 0.6 0.8 1.0
1.6
2.0
2.4
a
q
0
1
2
3
4
0.2 0.4 0.6 0.8 1.0
0e+0
03e
−08
6e−0
8
am
u
012
3
4
0.2 0.4 0.6 0.8 1.0
1.5
2.5
3.5
a
info
rmat
ion
gain
0
1
2
3
4
Figure 4: Scatterplots of estimated ETAS parameters as a function of estimates of a, for each
zone. The numbers in the plots indicate the corresponding zone numbers. Zone 1: active
continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,
zone 0: plate-interior.
44
![Page 45: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/45.jpg)
(a) .
5 6 7 8 9 10
05
1015
2025
magnitude
no. o
f dire
ct a
fters
hock
s
Zone 1Zone 2Zone 3Zone 4Zone 0
(b) .
0 2 4 6 8 10
0.70
0.75
0.80
0.85
0.90
0.95
1.00
time (years)
prop
ortio
n of
dire
ct a
fters
hock
s
Zone 1Zone 2Zone 3Zone 4Zone 0
(c) .
0 200 400 600 800
0.70
0.75
0.80
0.85
0.90
0.95
1.00
distance (km)
prop
ortio
n of
dire
ct a
fters
hock
s
Zone 1Zone 2Zone 3Zone 4Zone 0
(d) .
5.0 5.5 6.0 6.5 7.0 7.5 8.0
5010
015
020
025
0
magnitude
km fo
r 95
% o
f dire
ct a
fters
hock
s
Zone 1Zone 2Zone 3Zone 4Zone 0
Figure 5: Summaries of the estimated triggering functions for each zone. (a) The expected
number of direct (1st generation) aftershocks of an earthquake of a given magnitude, as a
function of that magnitude, according to the fitted ETAS model for each zone. (b) The
expected time required to contain a given proportion of direct aftershocks, according to the
fitted ETAS model for each zone. (c) The expected spatial range (in km) required to contain
a given proportion of direct aftershocks of a typical M6 event, according to the fitted ETAS
models. (d) The radius (in km) of a range expected to contain 95% of the direct aftershocks
of a typical earthquake of a given magnitude, as a function of that magnitude, according
to the fitted ETAS model for each zone. Zone 1: active continent, zone 2: slow-spreading
ridges, zone 3: fast-spreading ridges, zone 4: trenches, zone 0: plate-interior.
45
![Page 46: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/46.jpg)
(a) .
−122 −120 −118 −116
3233
3435
36
longitude
latit
ude
−9
−8
−7
−6
(b) .
−47 −46 −45 −44 −43
2022
2426
28
longitude
latit
ude
−9.0
−8.5
−8.0
−7.5
−7.0
−6.5
−6.0
−5.5
(c) .
145 146 147 148 149
−60
−59
−58
−57
−56
longitude
latit
ude
−8.5
−8.0
−7.5
−7.0
−6.5
−6.0
(d) .
140 141 142 143 144
3032
3436
38
longitude
latit
ude
−8.5
−8.0
−7.5
−7.0
−6.5
−6.0
Figure 6: Estimated conditional intensities according to ETAS (in color, on logarithmic
scale) and recorded earthquakes (black dots) for particular regions. The events are shallow
events (≤ 70 km deep) of magnitude ≥ 5.0. (a) A sub-region of zone 1 (active continent) in
California with 122 events in 1973-2006. (b) A sub-region of zone 2 (slow-spreading ridges),
an oceanic ridge in the North Atlantic Ocean, with 113 events in 1973-2006. (c) A sub-region
of zone 3 (fast-spreading ridges), an oceanic ridge South of Australia, containing 86 events
in 1973-2006. (d) A sub-region of zone 4 (trenches), East of Honshu, Japan, containing 185
events in 1999-2006.
46
![Page 47: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/47.jpg)
(a) .
radius(GCD in km)
Fre
quen
cy
0 50 100 150 200 250
010
2030
(b) .
−200 −100 0 100 200
−20
00
100
200
E−W direction (km)
N−S
dire
ctio
n (k
m)
(c) .
radius(GCD in km)
Fre
quen
cy
0 50 100 150 200 250
010
2030
40
(d) .
−200 −100 0 100 200
−20
00
100
200
E−W direction (km)
N−S
dire
ctio
n (k
m)
(e) .
radius(GCD in km)
Fre
quen
cy
0 50 100 150 200 250
010
2030
4050
60
(f) .
−200 −100 0 100 200
−20
00
100
200
E−W direction (km)
N−S
dire
ctio
n (k
m)
Figure 7: Histograms (left) and scatterplots (right) of events within 30 days and occurring
within the given great circle distance (GCD) of a given event of recorded magnitude. (a)
and (b): Magnitude 6.0 for zone 1, the active continent. (c) and (d): Magnitude 5.3 for zone
2, the slow-spreading ridge zone. (e) and (f): Magnitude 6.0 for zone 4, the trench zone.
47
![Page 48: Comparison of ETAS parameter estimates across di erent ...frederic/papers/annieetas1.pdf127 The ETAS model is a type of Hawkes point process model; such models are also sometimes 128](https://reader034.fdocuments.net/reader034/viewer/2022052010/60203dc14113e178316c5821/html5/thumbnails/48.jpg)
(a) .
0 50 100 150 200 250
0.00
0.02
0.04
0.06
0.08
0.10
radius (km)
(b) .
0 50 100 150 200 250
0.00
0.04
0.08
radius (km)
(c) .
0 50 100 150 200 250
0.00
0.02
0.04
0.06
0.08
radius (km)
(d) .
0 50 100 150 200 250
0.00
0.02
0.04
0.06
radius (km)
(e) .
0 50 100 150 200 250
0.00
0.04
0.08
0.12
radius (km)
Figure 8: Estimated triggering density g (solid curves) and empirical rate (dotted curves) of
events occurring within 30 days and within a given spatial radius, following an earthquake
of a given magnitude. (a) mainshocks of M6.0 in zone 1 (active continent). (b) mainshocks
of M5.3 in zone 2 (slow-spreading ridges). (c) mainshocks of M5.3 in zone 3 (fast-spreading
ridges). (d) mainshocks of M6.0 in zone 4 (trenches). (e) Mainshocks of M5.3 in zone 0
(plate-interior).
48