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Randomness of mega-thrust earthquakes implied by rapid stress recovery after the Japan M9 event



Constraining the recurrence of mega-thrust earthquakes is genuinely important for hazard assessment and mitigation. The prevailing approach to model such events relies on subduction zone segmentation and quasi-periodic recurrence due to constant tectonic loading. Here we analyze earthquakes recorded along a 1,000-km-long section of the subducting Pacific Plate beneath Japan since 1998. We find that the relative frequency of small to large events varies spatially, closely mirroring the large-scale tectonic regimes, and suggesting a laterally unsegmented mega-thrust interface. Starting some years before it broke, the Tohoku source region is imaged as a region of high stress concentration. Following the 2011 M9 earthquake, the size distribution changes significantly and most dramatic in the areas of highest slip. However, we discover that it returns within just a few years to its longer-term characteristics as observed prior to the mega-thrust event. This indicates a rapid recovery of stress and implies that such large earthquakes may not have a characteristic location, size or recurrence interval, and might therefore occur more randomly distributed in time.
Randomness of megathrust earthquakes
implied by rapid stress recovery after the
Japan earthquake
Thessa Tormann1*, Bogdan Enescu2, Jochen Woessner3and Stefan Wiemer4
Constraints on the recurrence times of subduction zone earthquakes are important for seismic hazard assessment and
mitigation. Models of such megathrust earthquakes often assume that subduction zones are segmented and earthquakes
occur quasi-periodically owing to constant tectonic loading. Here we analyse the occurrence of small earthquakes compared
to larger ones—the b-values—on a 1,000-km-long section of the subducting Pacific Plate beneath central and northern Japan
since 1998. We find that the b-values vary spatially and mirror the tectonic regime. For example, high b-values, indicative
of low stress, occur in locations characterized by deep magma chambers and low b-values, or high stress, occur where the
subducting and overriding plates are strongly coupled. There is no significant variation in the low b-values to suggest the
plate interface is segmented in a way that might limit potential ruptures. Parts of the plate interface that ruptured during the
2011 Tohoku-oki earthquake were highly stressed in the years leading up to the earthquake. Although the stress was largely
released during the 2011 rupture, we find that the stress levels quickly recovered to pre-quake levels within just a few years.
We conclude that large earthquakes may not have a characteristic location, size or recurrence interval, and might therefore
occur more randomly distributed in time.
Elastic rebound theory, introduced first by Reid following the
1906 San Francisco earthquake1, is one of the foundations
of earthquake science and explains how tectonic forces load
faults. It states that tectonic stresses build up on a fault over
decades, to be released within a major earthquake in seconds.
However, it is still unknown if this release is complete and followed
by a period of gradual reloading—and thus relative safety—or
if sufficient energy remains in the system to allow similar size
events more or less immediately. To explore this question, we
use a fundamental observation in seismology, the exponential
relationship between the frequency and magnitude of earthquakes,
known as Gutenberg–Richter law2, log10(N)=abM , where Nis
the number of events equal or above magnitude M, and aand bare
constants. This relationship is commonly used to infer occurrence
rates of infrequent large and hazardous events from the productivity
level (a-value) and size distribution (b-value) of abundant small-
to-moderate-magnitude seismicity. Although on a global average
b1, local b-values show substantial spatial variations—that is, in
some volumes the proportion of larger magnitudes is higher (b<1),
in others the proportion of small magnitudes exceeds the average
expectation (b>1).
Evidence from laboratory experiments3,4, numerical modelling5,
and natural seismicity6–8 indicates that b-values are negatively
correlated with differential stress. Fault patches of such-determined
significant stress accumulation have been observed to coincide with
locations of subsequent large earthquakes9,10. Low differential stress
conditions, for example, in high-pore-pressure regimes, lead to
high b-values, as observed in geothermal11 and volcanic6settings.
These observations suggest the use of b-values for mapping the
heterogeneous stress conditions in the Earth’s crust.
Spatial variation: time-invariant tectonic footprint
Our b-value study along the subducting Pacific plate off Japan is
the first of its kind to demonstrate how large-scale seismotectonics
imprint on the relative size distribution of earthquakes (for details
on the applied mapping, see Methods). We find the following
three major structural expressions, resolved with spatially varying
coverage, for any time period that we chose (Figs 1 and 2).
First, we observe a relatively homogeneous band of high b-values
at depths below 100 km following the volcanic front (Figs 1 and 2,
b>1.1). This structure represents the origin of the deep magmatic
root feeding the magma chambers of the volcanic chain6,12,13: dehy-
dration and partial melting of the subducting slab releases material
that ascends and eventually feeds the volcanoes above. Increasing
pore pressures reduce differential stresses and b-values increase,
as observed also during geothermal injection experiments11. The
b-values for crustal earthquakes below the volcanoes are equally
high, as shown along a cross-section at 40N (inset Fig. 1, b>1.1).
Latest volcanic activity/unrest along this cross-section has been
reported in the late 1990s (
Second, we image large volumes of low and very low b-values
from 100 km depth up to the trench (Figs 1 and 2, b<0.9).
The shallower part of the subduction interface is where the plates
are strongly coupled, although with significant depth and lateral
variations, and the accumulating stresses are released infrequently
by large or megathrust earthquakes14. Within this low-b-value
regime, we do not observe any significant segmentation that
would suggest inherent limits to potential ruptures. From the short
available observation period for estimating the depth extent of
co-seismic rupture from large subduction zone earthquakes, at
depths down to about 60km (ref. 15). We image low b-values down
1ETH Zurich, Swiss Seismological Service, Sonneggstrasse 5, Zürich 8092, Switzerland. 2University of Tsukuba, Faculty of Life and Environmental Sciences,
1-1-1 Tennodai, Tsukuba, Ibaraki 305-8572, Japan. 3Risk Management Solutions Inc., Stampfenbachstrasse 85, Zürich 8006, Switzerland. 4ETH Zurich,
Swiss Seismological Service, Sonneggstrasse 5, Zürich 8092, Switzerland. *e-mail:
© 2015 Macmillan Publishers Limited. All rights reserved
−200 km
−100 km
20 m 1.5
40 m
Cum. DEW number
Magnitude 8
Depth (km)
Depth (km)
Latitude (° N)
Longitude (° E)
138 140 142 144 146
40° N40° N
Longitude (° E)
Partial melt
Figure 1 | Three-dimensional study overview. Pre-Tohoku-oki (T2) b-values resolved along the subduction interface: high b-values in the low-stress
normal-faulting o-trench area; low and very low b-values along the megathrust. Star: 2011 M9 epicentre. White contours: slip model22 indicating the
M9 highest slip area (previously locked, highly stressed asperity, reflected by very low b-values); high b-values in the magma source region at depth
(dehydration and partial melting). Triangles: volcanoes. Insets: frequency–magnitude distributions (FMDs) for annotated nodes (A, B, C and D) from
dierent tectonic regimes, and cross-sectional view at 40N, as indicated, showing the relation between b-values for subduction and crustal seismicity
and surface volcanism. DEW, distance exponential weight.
to 100 km, which is consistent with the suggested extent of a
further deep coupled zone of high-stress accumulation beneath
Japan16. Thermal conditions would suggest that stresses here are
predominately released aseismically, possibly in decades-spanning
slip episodes following shallow megathrust events16. However, we
have no firm insight from our analysis if this is indeed the case, or if
the area can also rupture co-seismically in rare large events.
Third, along the outer rise, the off-trench region that has no
continental crust sitting atop exhibits lower differential stresses
typical for a shallow normal-faulting regime17. This is reflected by
comparatively low seismicity rates and high b-values (b>1.1). The
precise delineation of the steep b-value gradient from low to high
remarkably follows the trench line.
Temporal variation: asperity loading and unloading
Considering different periods, we find time-dependent signals that
are consistent with tectonic stress accumulation and release. We
select four periods that separate data before, in between, and after
very large events (Fig. 2): T1—before the rupture of the Tokachi-oki
M8 event, T2—three months after Tokachi-oki up to the Tohoku-
oki M9 event, T3—three months of Tohoku-oki aftershocks, and
T4—2013 onwards.
We find that the 2003 M8 Tokachi-oki event did occur within the
large and persistent low-b-value structure off Hokkaido18. (Fig. 2T1
and T2), but not in a distinct region of specifically low b-values.
The local b-values in the high-slip area increased only slightly in
the aftermath, and returned to the pre-mainshock level and lower
within 1–2 years (Figs 2T2 and 3). This suggests that although
an M8 event, the earthquake did not release a significant amount
of the overall stress that is continuously accumulating along that
part of the Pacific plate on a large enough area to host megathrust
earthquakes19. Although not an intuitive finding, this is consistent
with geodetic observations that reflect only minor fluctuations in
the subsidence rate related to M7–8 earthquakes over the past
120 years20, and independent results from co-seismic stress rotation
analysis, which concluded a <1% release of the background stress
through the M8 Tokachi-oki event21. For the Tohoku-oki event,
in contrast, the same study found strong stress rotations between
pre- and post-mainshock events and estimated a stress release of
>80% (ref. 21), which is equally reflected in the b-value results
discussed below.
As previously suggested10, we resolve a distinct low-b-value
structure in the subsequent high-slip area of the Tohoku-oki
mainshock, indicating locally a specifically strong stress
accumulation (Figs 1 and 2T2). This asperity on the plate
interface extends about 200km north–south and 100 km east–west,
reaching b-values of less than 0.5. This pattern was not visible
before 2003, and seems to have formed over a number of years10
(Figs 2T1 and 3). Similarly to laboratory observations of low and
decreasing b-values that could previously be detected as a fault of a
few centimetres length approached failure4, we find this for natural
earthquakes with fault sizes of hundreds of kilometres. However,
the timing of this long-term precursory signal remains unexplained,
as well as the observation that some low-b-value patches emerge
and subside without the occurrence of significant ruptures. Because
large earthquakes, for example, the 2003 Tokachi-oki event, do
not necessarily occur in regions of extremely low b-value either,
we cannot yet make conclusions about the quantitative predictive
power of b-value mapping.
In Figs 1 and 2, we show the spatial correlation of low pre-
mainshock b-values and high subsequent slip with respect to the
Yagi and Fukahata slip model22. As demonstrated in Fig. 4, this
trend of highest slip in the lowest-b-value regions is not sensitive
to the chosen slip model; we tested four more slip models23–26, and
all confirm the trend. Because largest slip would be expected in
volumes of highest previous stress and highest slip deficit, this low-
b–high-slip correlation provides another strong piece of evidence
that low-b-value structures can be usefully interpreted as mapping
asperities9—that is, largely locked, hard-to-break segments that can
sustain high levels of stress and tend to release the accumulated slip
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140 140145 145
140 140145 145
Longitude (° E)Longitude (° E)
Longitude (° E)Longitude (° E)
Latitude (° N)
Latitude (° N)
Latitude (° N)
Latitude (° N)
2 m
T1 Pre Tokachi-oki
1998 01 01−2003 09 26
−200 km
100 km
2 m
20 m
T2 Pre Tohoku-oki
2004 01 01−2011 03 11
−200 km
−100 km
20 m
T3 3 months aftershocks
2011 03 26−2011 06 30
200 km
−100 km
20 m
T4 Recent
2013 01 01−2014 09 12
−200 km
−100 km
0.5 1.0 1.5
Figure 2 | Temporal evolution of b-values along the subducting plate. Mapview along the three-dimensional slab structure for time periods T1–T4. Grey
nodes: nonlinear FMDs (ref. 7), stars: 2003 M8 Tokachi-oki and 2011 M9 Tohoku-oki epicentres, white contours: mainshock slip models, 2 m and 4 m for
Tokachi-oki49, 5–50 m for Tohoku-oki22. The maps show the first-order, time-independent tectonic imprint on b-values and the second-order temporal
variation imposed by the M9 mainshock. We note that the apparent shift in the depth of the high-b-value band between T1 and T2 might possibly be an
artefact caused by network and depth determination changes at that time.
deficit in large ruptures. A different terminology for this observation
is that low b-values indicate areas of strong coupling—for example,
reported along the Cocos Plate subducting beneath Costa Rica27.
Strongly coupled zones have also been suggested for the Tohoku area
pre-2011 (ref. 14), consistent with the observed low b-values.
Apart from the low b-values before the rupture, we find that
areas of large co-seismic slip during the Tohoku-oki event exhibit
a significant increase in b-values after the mainshock, representing
a strong stress release in these areas21,28 (Figs 2T2 and T3 and 3).
Again, the different slip models are consistent with respect to this
property (Fig. 4).
We find that b-values north and south from the major rupture
area are still persistently low after the Tohoku-oki event, and partly
even decreased (Figs 2T3 and T4). This is consistent with the stress
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Monthly rate b-value
2010 2015
2000 2005
2010 2015
Figure 3 |b-value time series and monthly activity rates. a,b-value and
uncertainty48 for the Tokachi-oki and Tohoku-oki earthquakes (red dashed
vertical lines) (for sampling parameters see Methods). b,M3events per
month inside the Tohoku-oki 10-m-slip contour22, red dashed line indicates
the average rate for 1998–2011.
transfer from the major event29; in particular, the analysis suggests
constant or even increased stress accumulation towards the north,
offshore Hokkaido, where the potential for megathrust events has
been repeatedly suggested19. We also map large areas of persistently
low b-values further south, offshore and beneath the greater Tokyo
area, which has experienced significant seismicity rate increases
Furthermore, apart from one small region in the northern half
of the pre-Tohoku lowest-b-value asperity, the high aftershock
b-values in all of the greater mainshock rupture area and down-slab
decrease significantly following the first few months of aftershocks
(Figs 2T3 and T4 and Supplementary Fig. 3). Assessed within
the 10-m-slip contour, the average b-value increase between T1
(b10m_T1 =0.81±0.02) and T3 (b10m_T3 =1.08 ±0.04) has been
recovered by 77% in T4 (b10m_T4 =0.87 ±0.03). This might seem
surprising on such a short timescale, as it indicates that stress
conditions are noticeably rebuilding in the greater asperity area
within three years from the M9 earthquake. However, the inference
of a rapid stress recovery is consistent with a similarly interpreted
observation from post-mainshock stress rotations, which suggest a
reloading of the asperity on the order of 6% of the stress drop within
the first eight months21.
To better understand the medium-term impact of the Tohoku-
oki event on b-values, we assess the observed changes between
pre-Tohoku-oki and 2013 onwards (between T1 +T2 and T4),
and compare those changes with the b-value differences observed
between pairs of two-year-long subsets of the catalogue—that is,
periods that are not dominated by large earthquake sequences. We
add to this comparison the strong changes observed between pre-
Tohoku-oki and the first three months of aftershocks (T1 +T2
versus T3), and also the impact of the Tokachi-oki event (Fig. 5).
The range and frequency of observed b-value changes between
the different periods (Fig. 5) confirm the partly unexpected
conclusions: spatio-temporal fluctuations in b-values are found to
be common in all two-year reference periods, and the impact of
the Tokachi-oki event is only of the order of that ‘background
fluctuation’—that is, it did not alter the stress field significantly; as
expected from the above analysis, there is a strong immediate impact
of the Tohoku-oki event; and the changes that we document for
the 2013-onward period have already significantly subsided towards
background levels, with the patch of remaining elevated b-values
in the northern half of the Tohoku asperity showing up as higher
amplitudes for large b-value increases compared to the ‘background
fluctuation’ (Fig. 5).
Potentially underlying physical processes
The b-values at the end of the study period have returned to values
similar to those seen between 1998 and 2003 (Figs 2, 3 and 5). This
recovery occurs in the same period of time it takes the aftershock
rate to reach the average long-term level (Fig. 3). Together these
imply a relatively rapid return over a few years to long-term
stationary loading conditions. Even the high tectonic loading rates
along this plate boundary alone are probably insufficient to explain
such rapid early stress recovery. However, observations of significant
coastal uplift20 post-Tohoku-oki indeed suggest that the proposed
deep coupled zone16 might have started moving aseismically after
the megathrust event, possibly contributing to reloading the asperity
above21. Because the processes at depth are still poorly understood,
and post-seismic strength and stress recovery are likely to be locally
complex and nonlinear, it would be dangerous to extrapolate the
b-value trend linearly into the future. It is speculative whether the
remaining stress difference will indeed be recovered within a few
years, as suggested by the present trend, or will take considerably
longer to rebuild.
It remains uncertain if the b-value recovery translates linearly
into a stress recovery of the same order3. Indeed, the correlation
between b-value and stress as observed in laboratory experiments3,4,
and indicated by this and previous seismicity studies6–8 , is not
unique: what drives the increase of the relative frequency of
large events in a particular area—that is, the likelihood that
small asperities preferably break together, rather than one by
one7,15? Whereas the stress level is an intuitive key parameter,
other different factors have been suggested to play a role, such
as the degree of material heterogeneity31, or the degree of stress
concentration (proportional to the product of stress and the square
root of the length of the nucleating fracture)32. Massive ruptures
and subsequent healing and coupling mechanisms might change
structural properties locally—that is, close to the main rupture
plane. It is difficult, however, to imagine how the structure changes
back and forth over larger volumes on the timescales on which
b-value changes have been documented. Overall, structural and
material properties at depth are even more difficult to infer and
confirm than stress conditions, and both might to some degree be
coupled and depend on each other.
Physical modelling might help to investigate and constrain what
components of the stress field imprint on the b-values most strongly.
The influence of the stress amplitude could be coupled to the degree
of homogenization or correlation of the stress field: neighbouring
locked patches could possibly move together, producing larger
magnitudes, if they are tied by stresses of the same order and
direction. Numerical modelling predicts that as a system approaches
system-level failure, individual ruptures become progressively more
correlated, meaning events occur closer together and grow larger,
thus the b-value decreases5. With the high convergence rate along
this plate boundary, such harmonization of the stress field, as
part of a fault healing mechanism, might be possible on short
timescales, and could be the dominant physical process behind
temporal b-value variation.
Inference on megathrust recurrence models
Our results indicate that the imprint of the Tohoku-oki event
strongly affects the earthquake size distribution, but for a
surprisingly limited period. The b-value and stress heterogeneity21
recovery are supplemented by an equally quick decay of seismicity
rates, which by mid-2014 have returned to <50% above the
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Ref. 22
Refs 23−26
Ref. 22
Refs 23−26
b-value (T2)
b-slip correlation
Δb-slip correlation
Di. pre/post Tohoku-oki
Δb (T3 − T2)
Slip (m)
20 40
Slip (m)
Figure 4 | Correlation between b-values and co-seismic slip in Tohoku. b-values estimated in 10-m slip bins, overlapping by 5 m, comparing dierent slip
models. Black line and grey shading: mean and standard deviation obtained for the slip model of ref. 22 shown in Figs 1 and 2, dark grey lines: mean for four
alternative tested slip models23–26.a, Pre-Tohoku-oki (T2) b-values compared with subsequent slip values, high slip was observed in areas of low b-values.
b, Correlation between slip and changes in b-value after the Tohoku-oki event (T3 versus T2), regions of high slip show significant increase in b-value.
T3 versus T1 + T2
T4 versus T1 + T2
Reference periods
−100 0 100
Δb (%)
Figure 5 | Significance of temporal b-value changes. Histograms of
observed b-value changes between dierent time periods. Cyan: T4 versus
T1 +T2, magenta: T3 versus T1 +T2, dark grey: two years post- versus
pre-Tokachi-oki (without first three months of aftershocks), black line and
grey shading: mean and standard deviation of changes observed between
six pairs of consecutive two-year periods that are not dominated by large
earthquakes (±2 years from 2000, 2001, 2006, 2007, 2008 and 2009).
background rate within the 10-m-slip area (Fig. 3). This is fully
consistent with the theoretically expected and observed anti-
correlation between aftershock duration and tectonic loading
rates: the faster the loading, the shorter the period of aftershock
activity17,33. Although the stress recovery process is probably
heterogeneous in space, and pockets of particularly high or low
stress30 might exist below the resolution of our b-value imaging,
we see no evidence for a lasting large-scale low-stress regime along
the ruptured plate interface, such as would be expected from the
seismic gap hypothesis34. Specifically, we hypothesize that the
Tohoku-oki event might be neither characteristic in location and
size, nor in a temporal sense.
Our b-value analysis suggests that, although with some degree
of local variation and temporal fluctuation, this megathrust zone
is more or less constantly and everywhere highly stressed, possi-
bly ready for large earthquakes any time with a low but on aver-
age constant probability35. We also find no indication for a long-
assumed lateral segmentation of the megathrust plate interface
into smaller areas that would only rupture in isolation and limit
the maximum magnitude36. This absence of identified barriers is
consistent with the occurrence of the Tohoku-oki event, with its
much larger magnitude than anticipated by some37,38. The resolved
b-values thus suggest that future ruptures may involve variable por-
tions of the megathrust, possibly overlapping with the Tohoku-oki
rupture plane, and equally likely extend towards the north or south.
Consequently, our results suggest that the renewal process along
this subduction zone is better described by a stationary Poissonian
process rather than a characteristic, partially time-predictable one39:
a similar size megathrust event is potentially possible in overlapping
volumes sooner than expected from estimated mean inter-event
times of past events40, whose variability along the Pacific plate is
large22,40, and possibly supporting a random occurrence hypothesis.
A longer palaeoseismic record along the Cascadia subduction
zone showed no evidence for characteristic recurrence either, but
suggested a clustered recurrence model41. Such behaviour would be
equally consistent with the results of our analysis, and could explain
how the long-term slip budget is balanced, as the high convergence
rate (8 cm yr1,>1,100 yr interval) suggests a still significant slip
deficit along most of the ruptured plate even post-Tohoku-oki.
A non-characteristic recurrence hypothesis for megathrust
events on the Pacific plate is in accordance with an independent
laboratory study on the recurrence behaviour of megathrust
events in two types of subduction zones42: for a model with
down dip segmentation, where ruptures cannot reach the trench,
characteristic ruptures evolve, whereas on an unsegmented interface
the evolving events occur randomly.
In prospect, both the first-order spatial imprint of the large-
scale tectonics and the transient changes in b-values imposed by
the M9 mainshock provide substantial information on the stress
field evolution that is at present not considered when evaluating
seismic hazard30. This could be directly integrated to improve future
generations of probabilistic seismic hazard assessment.
To study the spatial variation of relative stress conditions along the subduction
interface, we analyse b-values using the Japan Meteorological Agency (JMA)
earthquake catalogue along a roughly 1,000 km-long stretch of the Japanese
Pacific plate, following the three-dimensional geometry of the subducting
slab43 (Fig. 1).
Completeness magnitude, Mc.b-value analysis is critically dependent on a
robust estimate of completeness of the processed earthquake data. In particular,
underestimates in Mclead to systematic underestimates in b-values44. As
discussed in other studies45,Mcof the JMA earthquake catalogue improved
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significantly from 1998, when JMA started processing earthquake data recorded
by other Japanese institutions. We processed nearly 320,000 earthquakes, starting
in 1998, with M2.0, and assessed the temporal and spatial history of Mc(x,y,t)
locally at each grid node (Supplementary Fig. 1). Mcvaries locally and through
time, and very strongly early during aftershock sequences of large earthquakes.
We therefore did not consider the first two weeks and three months of
Tohoku-oki and Tokachi-oki aftershocks, respectively. For each of the time
periods we used a general cutoff (2.0 before Tohoku-oki, 3.7 for the aftershocks,
and 2.0 since 2013). We then estimated Mclocally for each node of our
2 km-spaced grid using the maximum curvature criterion6and added an extra 0.2
(ref. 44; Supplementary Fig. 1). Sensitivity tests showed that the interpreted
structures are stable over a wide range of cutoff magnitudes.
DEW sampling parameters and b-value estimation. Because the local stress
field primarily influences nearby earthquakes and should have a decaying impact
as a function of distance from a considered location, our b-value mapping uses
distance exponential weight (DEW) sampling7, which assigns each earthquake an
exponentially decaying, distance-dependent weight, w, according to its distance,
r, from the considered grid node: w(r)=0.7e0.07r(for example, an earthquake at
1 km distance has a weight wof 0.65, the correlation length (that is, where half
the maximum weight applies) is 10km, and at 75 km w=0.003). In such a way,
the events that are closest to a grid node gain the highest weight in mapping the
local size distribution, whereas distant events are considered with less importance.
This technique focuses the resolution on the plate interface, while sampling both
interface and intra-slab seismicity that may influence the local stress field. It
furthermore reduces the strong smoothing effect of the commonly applied
constant radius sampling and improves the resolution of local structures46.
We note that, as with any seismicity sampling parameter, the choice of the
decay parameter is slightly arbitrary46. However, it has been shown to clearly
resolve strong b-value anomalies for a crustal along-fault setting7,46, and we
verified that the interpreted structures in this study are insensitive with respect to
different parameter values. One property of this sampling technique and
parameter choice is its robustness to the choice of maximum radius
(Supplementary Fig. 2).
We sample earthquakes from a spherical volume of 75 km radius around each
grid node and calculate local b-values using the maximum likelihood estimate47
from the weighted frequency–magnitude distribution (FMD; ref. 7). We require
50 or more events above Mc, at least one event located closer than 25 km, and we
use a maximum of the 500 closest events per node. The standard deviation48 of
b-values decreases with the number of events, ranging from 15% for 50 events
to 5% for 500 events46. We interpret only the b-value patterns that are
significant well beyond the associated uncertainties, and we exclude from our
b-value analysis nodes for which the local FMD does not follow a single
exponential law7(grey areas in Fig. 2).
For time-series analysis for Tokachi-oki and Tohoku-oki, we use events inside
2 m (ref. 49) and 10 m (ref. 22) slip contours, in the depth intervals 20–100 km
and 0–30 km, respectively. Values are calculated from a moving window of 100
and 250 events, respectively, going event by event through the catalogue cut at
M2.0 and M3.0, respectively, applying maximum curvature6and adding 0.2
(ref. 44) to assess Mcin each time bin, always using >50 events. We note that
other than the spatial structures, which are robust with respect to the choice of
sampling parameters, the time series are rather sensitive and to be interpreted
with caution, and only in direct relation to and supplementing the spatial
analysis50. Choosing one-step sampling has the disadvantage of producing
maximum temporal correlations between neighbouring data points, so they
cannot be regarded as independent in further quantitative analysis or inference.
However, this sampling ensures that the time series exhibits all fluctuation
present in the data—that is, trends do not depend on the arbitrary choice of start
time of the analysis, as they would for larger step sizes, which randomly omit or
accentuate features/peaks.
Received 25 June 2014; accepted 16 December 2014;
published online 3 February 2015
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We thank J. Hardebeck, S. Jónsson and M. Wyss for feedback on the manuscript. We
thank JMA for sharing the earthquake catalogue. Figures were produced with The
Generic Mapping Tools Part of this study was funded
through SNF grant PMPDP2 134174. B.E. acknowledges support from the
‘Mega-Earthquake Risk Management’ project at the University of Tsukuba.
Author contributions
B.E. obtained, selected and pre-processed the earthquake data sets used in this study and
provided expert opinion on the Japanese seismotectonics. T.T. led the design of,
implemented and conducted the data analysis, and was responsible for result
visualization. S.W. and J.W. contributed to the design of the analysis. J.W. contributed to
figure generation. All authors participated in the discussion and interpretation of results
and the writing of the manuscript.
Additional information
Supplementary information is available in the online version of the paper. Reprints and
permissions information is available online at
Correspondence and requests for materials should be addressed to T.T.
Competing financial interests
The authors declare no competing financial interests.
© 2015 Macmillan Publishers Limited. All rights reserved
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We investigate the approach to catastrophic failure in a model porous granular material undergoing uniaxial compression. A discrete element computational model is used to simulate both the microstructure of the material and the complex dynamics and feedbacks involved in local fracturing and the production of crackling noise. Under strain-controlled loading, microcracks initially nucleate in an uncorrelated way all over the sample. As loading proceeds the damage localizes into a narrow damage band inclined at 30^{∘}-45^{∘} to the load direction. Inside the damage band the material is crushed into a poorly sorted mixture of mainly fine powder hosting some larger fragments. The mass probability density distribution of particles in the damage zone is a power law of exponent 2.1, similar to a value of 1.87 inferred from observations of the length distribution of wear products (gouge) in natural and laboratory faults. Dynamic bursts of radiated energy, analogous to acoustic emissions observed in laboratory experiments on porous sedimentary rocks, are identified as correlated trails or cascades of local ruptures that emerge from the stress redistribution process. As the system approaches macroscopic failure consecutive bursts become progressively more correlated. Their size distribution is also a power law, with an equivalent Gutenberg-Richter b value of 1.22 averaged over the whole test, ranging from 3 to 0.5 at the time of failure, all similar to those observed in laboratory tests on granular sandstone samples. The formation of the damage band itself is marked by a decrease in the average distance between consecutive bursts and an emergent power-law correlation integral of event locations with a correlation dimension of 2.55, also similar to those observed in the laboratory (between 2.75 and 2.25).
The 2011 Tohoku-oki earthquake caused large east- ward displacement and subsidence along the Pacific coast of northeastern Japan. This earthquake partly solved a well-known paradox holding that sense and rate of deformation differ greatly between geologic and geodetic estimates. A paradox remains, however, in explaining long-term uplift along the Pacific coast on a geologic time-scale. Geodetic data show that coastal subsidence continued at a nearly constant rate of ~5 mm/yr with small fluctuations associated with M7-8 interplate earthquakes for ~120 years before the Tohoku-oki earthquake. In an area near the Oshika Peninsula where coseismic subsidence is largest, extrapolation of a logarithmic function fitting observed postseismic deformation suggests that coseismic subsidence may be compensated for by the postseismic uplift for several decades but it is difficult to expect the postseismic uplift exceeding 2 meters, so it is implausible that the observed rapid subsidence continued throughout an entire interseismic period in a great megathrust earthquake cycle. We propose a hypothetical model in which the sense of vertical deformation changes from uplift to subsidence during the interseismic period. Using simple elastic dislocation theory, this model is explained by the shallow coupled part of a plate interface in an early interseismic period and the deep coupled part of a late interseismic period.
Monte Carlo magnitude mapping has been proposed as a tool to homogenize earthquake catalogs in time to overcome breaks in homogeneity of reporting. We analyze the uncertainty propagation for the b value due to that transformation and find that the additional error in b is negligibly small in relation to its standard deviation. However, in order to take the arbitrariness out of the proposed simulation, we argue that it has to be repeated a sufficient number of times to be stable, just like any other simulation technique. To illustrate this point, we show the example of cross-sectional mapping of the b value along the San Andreas fault near Parkfield, California.
Understanding and forecasting earthquake occurrences is presumably linked to understanding the stress distribution in the Earth's crust. This cannot be measured instrumentally with useful coverage. However, the size distribution of earthquakes, quantified by the Gutenberg-Richter b value, is possibly a proxy to differential stress conditions and could therewith act as a crude stress-meter wherever seismicity is observed. In this study, we improve the methodology of b value imaging for application to a high-resolution 3-D analysis of a complex fault network. In particular, we develop a distance-dependent sampling algorithm and introduce a linearity measure to restrict our output to those regions where the magnitude distribution strictly follows a power law. We assess the catalog completeness along the fault traces using the Bayesian Magnitude of Completeness method and systematically image b values for 243 major fault segments in California. We identify and report b value structures, revisiting previously published features, e.g., the Parkfield asperity, and documenting additional anomalies, e.g., along the San Andreas and Northridge faults. Combining local b values with local earthquake productivity rates, we derive probability maps for the annual potential of one or more M6 events as indicated by the microseismicity of the last three decades. We present a physical concept of how different stressing conditions along a fault surface may lead to b value variation and explain nonlinear frequency-magnitude distributions. Detailed spatial b value information and its physical interpretation can advance our understanding of earthquake occurrence and ideally lead to improved forecasting ability.
Goldfinger et al. (2012) interpreted a 10,000-year old sequence of deep sea turbidites at the Cascadia subduction zone (CSZ) as a record of clusters of plate-boundary great earthquakes separated by gaps of many hundreds of years. We performed statistical analyses on this inferred earthquake record to test the temporal clustering model and calculate time-dependent recurrence intervals and probabilities. We used a Monte Carlo simulation to determine if the turbidite recurrence intervals follow an exponential distribution consistent with a Poisson (memoryless) process. The latter was rejected at a statistical significance level of 0.05. We performed a “cluster analysis” on 20 randomly simulated catalogs of 18 events (event T2 excluded), using ages with uncertainties from the turbidite dataset. Results indicate 13 catalogs exhibit statistically significant clustering behavior, yielding a probability of clustering of 13/20 or 0.65. Most (70%) of the 20 catalogs contain 2 or 3 closed clusters (a sequence that contains the same or nearly the same number of events) and the current cluster T1 to T5 appears consistently in all catalogs. Analysis of the 13 catalogs that manifest clustering indicates that the probability that at least one more event will occur in the current cluster is 0.82. Given that the current cluster may not be closed yet, the probabilities of a M 9 earthquake during the next 50 and 100 years were estimated to be 0.17 and 0.25, respectively. We also analyzed the sensitivity of results to including event T2, whose status as a full-length rupture event is in doubt. The inclusion of T2 did not change the probability of clustering behavior in the CSZ turbidite data, but did significantly reduce the probability that the current cluster would extend to one more event. Based on the statistical analysis, time-independent and time-dependent recurrence intervals were calculated.
The statistics of large earthquakes commonly involve large uncertainties due to the lack of long-term, robust earthquake recordings. Small-scale seismic events are abundant and can be used to examine variations in fault structure and stress. We report on the connection between stress and microseismic event statistics prior to the possibly smallest earthquakes: those generated in the laboratory. We investigate variations in seismic b value of acoustic emission events during the stress buildup and release on laboratory-created fault zones. We show that b values mirror periodic stress changes that occur during series of stick-slip events, and are correlated with stress over many seismic cycles. Moreover, the amount of b value increase associated with slip events indicates the extent of the corresponding stress drop. Consequently, b value variations can be used to approximate the stress state on a fault: a possible tool for the advancement of time-dependent seismic hazard assessment.