Öncel Akademi: İstatistiksel Sismoloji

Short Note
Is There a Basis for Preferring Characteristic Earthquakes over
a Gutenberg–Richter Distribution in Probabilistic
Earthquake Forecasting?
by Tom Parsons and Eric L. Geist
Abstract The idea that faults rupture in repeated, characteristic earthquakes is cen-
tral to most probabilistic earthquake forecasts. The concept is elegant in its simplicity,
and if the same event has repeated itself multiple times in the past, we might anticipate
the next. In practice however, assembling a fault-segmented characteristic earthquake
rupture model can grow into a complex task laden with unquantified uncertainty.
We weigh the evidence that supports characteristic earthquakes against a potentially
simpler model made from extrapolation of a Gutenberg–Richter magnitude-frequency
law to individual fault zones. We find that the Gutenberg–Richter model satisfies key
data constraints used for earthquake forecasting equally well as a characteristic model.
Therefore, judicious use of instrumental and historical earthquake catalogs enables
large-earthquake-rate calculations with quantifiable uncertainty that should get at least
equal weighting in probabilistic forecasting.
Introduction
Shouldweexpectlarge(M ∼7andgreater)earthquakesto
behave differently than small ones? Probabilistic earthquake
forecasting is commonly driven by the idea that large, charac-
teristic earthquakes repeatedly rupture the same fault seg-
ments with the same magnitude and slip distribution,
departing from the expected magnitude-frequency trend of
smaller events. The characteristic model is derived from
paleoseismic observations (Schwartz and Coppersmith,
1984; Wesnousky, 1994; Hecker and Abrahamson, 2004)
and has great appeal for forecasting because if the same event
has occurred repeatedly in the past, there is implied predict-
ability. For the last ∼20 yr, global earthquake forecasts have
employed characteristic earthquake models to calculate ex-
pected rupture rates (e.g., McCann et al., 1979; Working
Group on California Earthquake Probabilities [WGCEP],
1988, 1990; Nishenko, 1991; WGCEP, 1995; Frankel et al.,
2002; WGCEP, 2003; Parsons, 2004; Earthquake Research
Committee, 2005; Romeo, 2005; Petersen et al., 2008;
WGCEP, 2008).
By definition, characteristic earthquakes on individual
faults are not subject to magnitude-frequency laws (Ishimoto
and Iida, 1939; Gutenberg and Richter, 1954) wherein earth-
quakes are observed to obey a linear power-law distribution
as log NM a bM, where a and b are constants, and N
is the number of earthquakes greater than magnitude M. That
distribution is commonly referred to as Gutenberg–Richter
behavior. For very large regions, there is no dispute that
earthquakes of all magnitudes scale according to the linear
power law. Gutenberg–Richter behavior is also observed for
subregions and individual faults for the most frequent lower-
magnitude earthquakes (e.g., Abercrombie and Brune, 1994;
Westerhaus et al., 2002; Schorlemmer et al., 2003, 2004;
Wyss et al., 2004; Parsons, 2007). However, because char-
acteristic earthquakes on a given fault are thought to always
be about the same size, their rates are proposed to be higher
than expected from extrapolation of a linear Gutenberg–
Richter relation (Fig. 1) (e.g., Schwartz and Coppersmith,
1984; Wesnousky, 1994; Hecker and Abrahamson, 2004;
c.f., Kagan, 1996).
Implications of a chosen earthquake distribution model
on earthquake probability calculations are critical. For a char-
acteristic model, much effort is exerted on seeking out earth-
quake return times of infrequent, specific large-magnitude
events that are expected to represent almost all the hazard.
With the characteristic model, it is difficult to quantify uncer-
tainties surrounding fault segmentation, characteristic magni-
tude assignment, and rupture boundaries (e.g., Savage, 1991,
1992; Page and Carlson, 2006). Often, these uncertainties are
omitted (e.g., WGCEP, 2008). If instead a linear Gutenberg–
Richter model applies to individual faults, then rates of the
largest events are directly extrapolated from observation of
smaller, instrumentally recorded earthquakes. A further
implication of a Gutenberg–Richter model is that ruptures
can occur over a broader magnitude range.
2012
Bulletin of the Seismological Society of America, Vol. 99, No. 3, pp. 2012–2019, June 2009, doi: 10.1785/0120080069

It is not uncommon for aspects of Gutenberg–Richter and
characteristic behavior to be incorporated in probabilistic
forecasting, although there is typically a strong preference to-
ward use of the characteristic model for major faults (e.g., type
A faults, which typically have available paleoseismic data and
highest slip rates [WGCEP, 1995]). As an example, character-
istic rupture models got 90% weighting for sources of the
largest possible earthquakes in the WGCEP (2008) forecast.
The fit of both characteristic and Gutenberg–Richter rupture
models to observations is examined here because probability
and shaking hazard calculations are so strongly affected by
model choice.
Earthquake Distribution Model and Data Constraints
Key data that inform earthquake rate forecasts are geo-
logic and geodetic estimates of long-term fault slip rates and
paleoseismic event rates and magnitudes, as well as instru-
mental/historical earthquake catalogs. As of this writing,
official forecasts in the United States do not yet embrace
physics-based models because of the added number of free
parameters (e.g., Petersen et al., 2008; WGCEP, 2008).
Instead, the preferred method has been to derive rupture
models from the aforementioned empirical constraints. By
definition, a characteristic earthquake model readily fits
the empirical data because (1) in the model, virtually all fault
slip is caused by repeated characteristic events that mimic the
long-term slip distribution, (2) the model is derived from
paleoseismic observations in the first place, and (3) if a large
enough region is considered, it can be fit to a regional
Gutenberg–Richter distribution, although this can be a chal-
lenge (c.f., WGCEP, 2008). Less certain is whether a model
with linear Gutenberg–Richter behavior occurring on indi-
vidual faults can meet these observed data constraints.
We developed an algorithm to invert for long-term earth-
quake distribution on theoretical faults to assess the use of a
Gutenberg–Richter model. Constraints were geologic slip
rate and a fixed number of earthquakes (magnitude range
M 5:0–7:8) obeying a linear Gutenberg–Richter distribution
with b-value (slope) of 1.0. Our example model had two
branching strike-slip fault surfaces 700 and 500 km long,
discretized into 2.5 by 2.5 km patches that slipped at 40
and 20 mm=yr, respectively, for 1000 yr (Fig. 2). Model
duration, slip rates, and rake are trade-offs that only matter
in terms of establishing seismic moment balance in the
model.
Initial earthquake nucleation sites were assigned
randomly, and magnitudes were sampled from the
Gutenberg–Richter distribution. Rupture areas were assigned
from the empirical magnitude-area relation of Hanks and
Bakun (2008). Event slip was calculated through the
moment-magnitude relation of Hanks and Kanamori
(1979) assuming a shear rigidity of μ 3 × 1010 N m.
Contiguous, uniform-slip ruptures were initially allowed
to grow at random into adjoining patches until they covered
their magnitude-appropriate areas. Iterations were updated
by assigning slip of a growing rupture preferentially to
patches with the smallest prior accumulated slip, with the
assumption that lowest prior slip implies highest failure
stress. As the model was updated, if a randomly assigned
hypocenter landed in a patch that produced cumulative slip
in excess of the cumulative long-term slip constraint (plus
10% uncertainty), that event was repeatedly moved to
new locations at random until its slip could be accommo-
dated. Ruptures were allowed to jump between fault
branches within a 4 km distance (Barka and Kandinsky-
Cade, 1988; Harris, 1992; Harris and Day, 1993, 1999;
Lettis et al., 2002), with the decision governed by preexisting
slip distribution (low slip equals high stress); modeled
dynamic rupture jumps are shown to be most encouraged
by a high stress state (e.g., Harris and Day, 1993, 1999).
Figure 2b shows that a Gutenberg–Richter model was able
to fit observed geologic slip rates within 3 mm=yr.
Calculated event rates for M ≥6:7 earthquakes averaged
about 7:5 2 per 1000 yr on the two segments, similar to
the observed average rate of 6 2 observable events per
1000 yr on comparable southern San Andreas fault segments,
which have long records (∼300–1500 yr) and slightly
slower slip rates (∼20–35 mm=yr) than the model faults
(Biasi et al., 2002; Fumal, Rymer, and Seitz, 2002; Fumal,
Weldon, et al., 2002).
Examination of Characteristic Earthquake
Model Constraints
Two primary arguments are made in support of the char-
acteristic earthquake model: (1) where repeated paleoearth-
quake offsets are measured, they are relatively uniform, and
(2) paleoseismic event rates are higher than expected from
extrapolation of a linear Gutenberg–Richter slope using
smaller earthquakes recorded on or near a fault (Schwartz
and Coppersmith, 1984; Wesnousky, 1994) (Fig. 1). We
examine each of these arguments with respect to a simulated
Magnitude
A: Linear Gutenberg-Richter B: Characteristic modelLog(N)
Paleo-
seismic
rate
Instrum
ental
m
entalrate
Figure 1. (a) Sketch of typical linear Gutenberg–Richter
magnitude-frequency relation. In (b), there is a bump where high-
er-rate characteristic large-magnitude earthquakes are proposed.
Short Note 2013

earthquake catalog built from the Gutenberg–Richter model
shown in Figure 2.
Event-Slip Observations
The characteristic earthquake model was born from
recognizing repeated, very similar fault offsets in the geo-
logical record, particularly along the Wasatch fault zone
in Utah, and the San Andreas fault in southern California
(Schwartz and Coppersmith, 1984) and reinforced globally
(Hecker and Abrahamson, 2004). Plotted in Figure 3 (blue
line and error bars) are observed individual offsets and un-
certainty for the Wasatch segments (Chang and Smith, 2002)
and the south San Andreas fault site of Wrightwood (Weldon
et al., 2004). The issue is complicated on the San Andreas
fault because the best event-slip measurements available are
from the Wrightwood site, which lies at an assigned segment
boundary (WGCEP, 2008) and may thus see ruptures from
two different characteristic segments, although the distribu-
tion of event slip is not bimodal (Fig. 3a). Dip-slip offsets
such as those along the Wasatch front are easier to measure
Figure 2. (a) Perspective view of the model fault configuration, contoured slip, and event-rate (M ≥6:7) participation for a Gutenberg–
Richter distribution on faults. (b) Histograms with the magnitude-frequency distribution as well as variation of model slip and events on 2.5
by 2.5 km patches.
2014 Short Note

in the paleoseismic record than strike-slip ruptures because
they leave visible fault scarps.
To test a Gutenberg–Richter model against event-slip
observations, we made repeated random draws of slip events
from surface points of the model depicted in Figure 2 to
simulate each of the sites plotted in Figure 3. Because the
model in Figure 2 is slip balanced, the time, rake, and
slip-rate factors cancel each other out, and we can use the
model to compare the spatial distribution of earthquake rup-
tures to any real fault site up to its maximum expected mag-
nitude threshold. We can thus use the Gutenberg–Richter
distribution of event slip drawn from the moment-balanced
1
2
3
Eventslip(m)Eventslip(m)
C: Wasatch - Brigham City
1
2
3
B: Wasatch - Multisegment ruptures
2
4
6
8
A: San Andreas - Wrightwood
1
2
3
D: Wasatch - Weber
1
2
3
4
Eventslip(m)
E: Wasatch - Salt Lake City
1
2
3
4
F: Wasatch - Provo
1
2
3
4
Eventslip(m)
G: Wasatch - Nephi
Event
Figure 3. Observed individual event slip from paleoearthquakes at (a) Wrightwood on the San Andreas fault (Weldon et al., 2004), and
(b–g) Wasatch fault segments (Chang and Smith, 2002). Blue lines show observed slip values in meters, blue error bars indicate measurement
uncertainty, and values are sorted from smallest to largest. Red lines show means of slip drawn from the Gutenberg–Richter model of Figure 2,
with red error bars showing 95% of the range of 50 random draws. We assumed, based on the observations, that the minimum threshold of
observation is ∼0:7 m (M ∼6:6) on the San Andreas fault and ∼1:0 m (M ∼6:8) on the Wasatch. (b) Mean slip values from the Wasatch
multisegment model of Chang and Smith (2002); (c–g) slip values if ruptures are confined to single segments.
Short Note 2015

model for comparison with both the San Andreas and
Wasatch fault observations. We assumed, based on the ob-
servations plotted in Figure 3, that 0.7 m offsets (M ∼6:6)
represent the smallest resolvable paleoslip on the San
Andreas fault and 1.0 m offsets (M ∼6:8) are resolvable
on the Wasatch fault. Given the nature of power-law magni-
tude-frequency distributions, the most likely ruptures to have
occurred in the model at a given site are the smallest ones
(Fig. 2b). Therefore, limited sampling (i.e., the 2–15 events
of Fig. 3) yields a series of individual slip events that have
almost the same offset.
When we randomly draw 50 sets of 2–15 synthetic
ruptures from the Gutenberg–Richter model of Figure 2,
we are able to reproduce observations of Figure 3 (red error
bars) within reported uncertainties (blue error bars) with the
exception of the Provo single-segment event-slip model
(Fig. 3f) on the Wasatch fault. Observations that are matched
by a Gutenberg–Richter model include San Andreas fault
offsets at Wrightwood (Weldon et al., 2004) (Fig. 3a), the
Wasatch fault multisegment rupture models of Chang and
Smith (2002) (Fig. 3b), and all but one of the Wasatch
single-segment rupture models. We conclude from this that
a Gutenberg–Richter model should not be excluded based on
observation of similar rupture offsets.
Instrumental versus Paleoseismic
Magnitude-Frequency Trends
For a broad region, magnitude-frequency resolution is
such that a mixture of characteristic faults (e.g., Wesnousky,
1994; López-Ruiz et al., 2004) or Gutenberg–Richter distrib-
uted faults with different maximum magnitude cutoffs (e.g.,
Kagan 2002a,b) can satisfy the linear power-law trend. The
characteristic-versus-power-law issue hinges on more poorly
resolved relations from individual fault zones. Arguments for
the characteristic earthquake model are supported by calcu-
lations of higher paleoseismically determined earthquake
rates than extrapolation of a power-law derived from earth-
quakes instrumentally recorded on and near individual faults
(Schwartz and Coppersmith, 1984; Wesnousky, 1994) (e.g.,
Fig. 1b).
We plot magnitude-frequency distributions (see the Data
and Resources section) in Figure 4a for a section of the
southern San Andreas fault (identified in Fig. 4b), where
there are numerous paleoseismic sites and abundant catalog
seismicity. Comparisons between paleoseismic and instru-
mental magnitude-frequency behavior is very much depen-
dent on both temporal and spatial sampling of the catalog
and how paleoseismic rates are calculated (e.g., Stein and
Newman, 2004; Stein et al., 2005; Parsons, 2008). The
paleoseismic rates we use for comparison in Figure 4 (from
Biasi et al., 2002; Fumal, Rymer, Seitz, 2002; Fumal,
Weldon, et al., 2002) were developed through uncertainty
analysis by Biasi et al. (2002) and Parsons (2008) as used
in the WGCEP (2008) earthquake rupture forecast, and the
magnitude range is from the Weldon et al. (2004) event-slip
data.
It is apparent that broader sampling of earthquake cat-
alogs either temporally or spatially increases the overall rate.
A very wide swath taken from 50 km on either side of the
southern San Andreas fault from the ∼1970–2007 catalog
(see the Data and Resources section) produces a Guten-
berg–Richter trend that is in accord with paleoseismic rates,
whereas a 5 km swath from the instrumental catalog
projects lower rates. Sampling a 5 km swath from a longer
∼1850–2006 earthquake catalog (Felzer and Cao, 2008)
(Fig. 4a) is also consistent with paleoseismic rates. The his-
toric catalog has relatively few events, 20 M >4 shocks were
located within 5 km of the San Andreas segment we
investigated (Fig. 4b) and is subject to significant location
uncertainty for the oldest events.
The instrumental catalog along the San Andreas fault
may be influenced by the legacy of past large earthquakes
(Harris and Simpson, 1996; Stein, 1999) or inherent rate
fluctuations (Felzer and Brodsky, 2005). However, a narrow
( 5 km) swath taken from a more active San Andreas fault
segment in the Parkfield region is also consistent with south-
ern San Andreas paleoseismic rates. The Parkfield segment
did not participate (Harris and Simpson, 1996; Thatcher
et al., 1997) in either of California’s largest historical earth-
quakes (the 1857 M ∼7:9 Fort Tejon and the 1906 M ∼7:8
San Francisco [e.g., Ellsworth, 1990] shocks), and thus
might not be subject to stress shadowing effects if they have
impacted seismicity rates (Harris and Simpson, 1996).
Attempting to reconcile paleoseismically determined
event rates with Gutenberg–Richter relations projected with
smaller, instrumental or historical earthquakes highlights the
broader questions of what constitutes a fault zone, and how
earthquakes interact with each other. If the analysis were re-
stricted to the exact fault surface that ruptures during the
largest earthquakes, it is clear that over instrumental catalog
durations of a few decades, Gutenberg–Richter projections
cannot match south San Andreas paleoseismic rates, imply-
ing support for a characteristic model (Fig. 1b). However,
indications are that a longer (∼150 yr) catalog from a imply-
ing narrow ( 5 km) swath can also match the paleoseismic
record (Fig. 4a).
Even the narrowest spatial definition ( 5 km wide) of
the San Andreas fault zone we consider incorporates minor
subsidiary faults along with the main trace (Fig. 4b). It is
clear that fault-zone definition, as well as earthquake catalog
completeness and temporal rate fluctuations are important
sources of uncertainty in applying a linear Gutenberg–
Richter model to individual fault zones. However, in
California and much of the rest of the world, the combined
instrumental and historical catalogs are such that magnitude-
frequency uncertainties are readily quantifiable (e.g., Felzer
and Cao, 2008). In summary, examination of instrumental
and historical earthquake catalogs shows that rates are de-
pendent on space and time; thus completeness issues prevent
us from identifying a clear basis for preferring either a char-
2016 Short Note

acteristic earthquake model or a linear Gutenberg–Richter
model on individual fault zones.
Conclusions
Probabilistic earthquake forecasts rely almost entirely
on a small set of empirical data that the characteristic earth-
quake model readily matches. In probabilistic forecasts like
the WGCEP (2008) effort, characteristic models get dominant
weight for sources of the largest earthquakes. To determine if
this tilt is justified, we compare the characteristic earthquake
model with another empirical approach, a Gutenberg–
Richter-distributed rupture model. We conclude that giving
clear preference to either model in calculating future earth-
quake rates is not justifiable because a fault-based, linear
Gutenberg–Richter distribution of earthquakes seems to
match available data just as well as a characteristic model.
The data we consider include the most-commonly used
forecast constraints, such as long-term geologically and
geodetically determined fault-slip rates, paleoseismic event
Figure 4. (a) Magnitude-frequency relations for a variety of earthquake catalogs, such as events taken from 5, 10, and 20 km of either side
of the south San Andreas fault (epicenters shown in [b]). Also shown are relations for events taken from 50 km of either side of the fault trace,
5 km either side of a 100 km long section of the San Andreas from the more active Parkfield region and events taken from 5 km of either side
of the south San Andreas fault from the 1850–2006 catalog of Felzer and Cao (2008) (red stars on [a] and [b]). Inset table gives individual
historic event data. Blue symbols show paleoseismic event rates for south San Andreas sites from Biasi et al. (2002) and Parsons (2008)
and cover rate and magnitude uncertainties as applied by WGCEP (2008). Paleoseismic magnitudes are unknown, so rates are displayed
between M 6:6 and M 8. Stress-renewal recurrence intervals for events between M 7 and M 8 from Parsons (2006) are shown by red
symbols.
Short Note 2017

rates, individual event-slip observations, and subregional
magnitude-frequency distributions. An important influence
on using a linear Gutenberg–Richter model for extrapolating
large earthquake rates is how a fault zone is defined in terms
of which smaller earthquakes can be associated with it.
However, rate uncertainties related to catalog selection are
readily quantifiable and transferable to forecast uncertainty,
which is in contrast with the often unwieldy and sometimes
unquantifiable uncertainties associated with characteristic
earthquake models.
Data and Resources
Earthquake catalogs used in this study to calculate mag-
nitude-frequency distributions were acquired through the
Advanced National Seismic System (ANSS) catalog search
linked through the Northern California Earthquake Data
Center (NCEDC) web site at http://www.ncedc.org/anss/
catalog‑search.html (last accessed April 2008).
Acknowledgments
We very much appreciate reviews by David Schwartz and Ivan Wong,
both of whom prefer the characteristic earthquake model, but nonetheless
supported publication of this alternative view, and provided many helpful
comments that improved this discussion.
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2018 Short Note

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U.S. Geological Survey
MS-999, 345 Middlefield Road
Menlo Park, California, 94025
Manuscript received 15 May 2008
Short Note 2019

Öncel Akademi: İstatistiksel Sismoloji

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