Tectonophysics 621 (2014) 198–210
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Tectonophysics
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Stress drop as a criterion to differentiate subduction zones where Mw 9
earthquakes can occur
Tetsuzo Seno ⁎
Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo 113-0032, Japan
a r t i c l e
i n f o
Article history:
Received 11 September 2013
Received in revised form 13 February 2014
Accepted 23 February 2014
Available online 11 March 2014
Keywords:
Stress drop
Mw 9 earthquake
Asperity
Scaling relation
Pore fluid pressure ratio
Seismic coupling
a b s t r a c t
I propose a hypothesis that might be used to differentiate zones that produce Mw ≥ 9 earthquakes from zones
that do not. I calculate stress drop (Δσ) values, for Mw ≥ 7 thrust-type earthquakes over worldwide subduction
zones, compiling the studies that obtained well-constrained slip distributions by inverting seismic, geodetic or
tsunami data. Earthquakes are grouped into class 1: Mw ≥ 9 earthquakes, class 2: Mw b 9 earthquakes in a subduction zone segment in which at least one Mw ≥ 9 earthquake has occurred, and class 3: earthquakes in a subduction zone segment in which no Mw ≥ 9 earthquake has occurred. A total of 53 earthquakes are analyzed. The
average stress drop (Δσ) values of the class 1, 2, and 3 events are 4.6, 3.4 and 1.6 MPa, respectively. In individual
subduction zones, Δσ values of the class 2 events are by more than twice greater than those of the class 3 events,
except Kuril–Hokkaido. Based on these results, I propose a hypothesis that if Δσ is greater than 3 MPa in a subduction zone segment, this segment possibly produces Mw ≥ 9 earthquakes, and if Δσ is less than 2 MPa, the segment would not produce Mw ≥ 9 earthquakes. I examine the fault parameters obtained in this study based on the
newly derived scaling relations that take into account the variation in Δσ. The rupture of subduction zone earthquakes, from Mw ≥ 9 through Mw ~ 7, can be understood on the basis of the same scale-invariant physics, if Mo
is normalized by Δσ1.5 in the scaling relation between L and Mo. Using this relation, I estimate the maximum
magnitude of an earthquake which may rupture the entire Nankai–Suruga Trough off SW Japan, and obtain
Mw = 8.6–8.4.
© 2014 Elsevier B.V. All rights reserved.
1. Introduction
On March 11, 2011, a Mw 9 interplate earthquake occurred in the
subduction zone east of northern Honshu (northern Honshu and its offshore region are denoted by Tohoku and Tohoku-oki, respectively, hereinafter). Twenty thousand people were killed, in part because the
earthquake was much bigger than anticipated. Although Seno (1979)
noted, more than 30 years ago, a possibility of the occurrence of a
Mw ~ 8 earthquake in the offshore region of Miyagi prefecture, the eastern part of middle Tohoku, the expected size was much smaller than
the 2011 Tohoku-oki event. More recently, the Headquarters for
Earthquake Research Promotion of the Japanese Government (2010)
had assigned an 80–90% probability to the occurrence of a Mw ~ 8.0
earthquake within 30 years far offshore Miyagi, with a magnitude similarly smaller. These lower estimates were made because the quantification on the level of the seismicity off Miyagi was based on the fact that
Mw ~ 8 earthquakes have historically occurred in the offshore region
from Hokkaido toward Miyagi (Kanamori, 1977; Lay and Kanamori,
1981; Seno, 1979; Uyeda and Kanamori, 1979).
⁎ Tel.: +81 3 5802 5747; fax: +81 3 5841 3391.
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.tecto.2014.02.016
0040-1951/© 2014 Elsevier B.V. All rights reserved.
On the other hand, the age and convergence rate of the subducting
plate were proposed as the controlling factors of the maximum magnitude in each subduction zone by Ruff and Kanamori (1980, 1983a). The
fact that the old Pacific plate converges at a fast rate in the Tohoku-oki
region had appeared to be consistent with the above quantification of
the seismicity. After the occurrence of the 2004 Sumatra–Andaman
and 2011 Tohoku-oki earthquakes, however, Stein and Okal (2007,
2011) noted that these factors did not give good estimates for the maximum magnitudes in these regions (see also Gutscher and Westbrook,
2009; Heuret et al., 2011). Before the Sumatra–Andaman event, it had
also been noted that they do not explain the variation of the seismic coupling ratio, that is, the seismic slip rate/plate motion, among the world's
subduction zones well (Pacheco et al., 1993; Peterson and Seno, 1984;
Scholz and Campos, 1995). After the Sumatra–Andaman event, some
workers proposed that Mw ≥ 9 earthquakes can occur even in any subduction zone; in particular, McCaffrey (2007) stated “For policy purposes, one lesson we should take away from the Sumatra–Andaman
earthquake is that every subduction zone is potentially locked, loaded,
and dangerous. To focus on some and ignore others may be folly”. The
occurrence of the 2011 Tohoku-oki earthquake seems to support this
statement.
The purpose of the present study is to examine whether this is
correct or not. In other words, I try to find an indicator, if any, to
T. Seno / Tectonophysics 621 (2014) 198–210
differentiate Mw ≥ 9 subduction zones from Mw b 9 subduction zones. I
show that the stress drop (Δσ) values of earthquakes in each subduction zone segment may give such an indicator. Δσ in this study is the
static one derived from a dislocation in the elastic medium under the
applied uniform shear stresses (Eshelby, 1957). It has been said that
Δσ values of interplate earthquakes are more or less constant on the
order of 0.1–1 MPa (e.g., Aki, 1972; Hanks, 1977; Kanamori and
Anderson, 1975; Purcaru and Berckhemer, 1982). Looking at Δσ values
more closely, however, a distinction in average stress drop (Δσ) between Mw ≥ 9 subduction zones and Mw b 9 subduction zones may be
found.
To understand the role of the variation in Δσ on the rupture process
of the subduction zone thrust-type earthquakes, I derive new scaling relations, with non-constant Δσ, between S and D, where S and D are the
fault area and the average dislocation, respectively, and between L and
Mo, where L and Mo are the fault length and the seismic moment, respectively. I apply the latter relation to predict the magnitude of an
earthquake that may rupture the entire Nankai–Suruga Trough. I also
show that Δσ should be proportional to 1 − λ; λ is the pore fluid pressure ratio defined by Pw = λσn, where Pw is the pore fluid pressure and
σn is the normal stress at the subduction zone megathrust. This implies
that 1 − λ could be used as a proxy for Δσ, and the variation of λ, which
is controlled by the hydration/dehydration reactions and the escape of
fluids to the surroundings over a geologic time, may cause the observed
variation of the seismic coupling; I use “seismic coupling” to represent
the mode of moment release in subduction zones more generally than
the seismic coupling ratio, that is, the seismic slip rate/plate motion.
2. Calculation of stress drops
The static stress drop Δσ associated with D is generally represented
by
Δσ ¼ CμD=l;
ð1Þ
where l is a representative fault dimension, μ is the rigidity, and C is a
constant that depends on the fault geometry. At most instances in this
paper, I use the stress drop representation for a thin elliptic crack
(Eshelby, 1957). Let the ellipsoid be represented as
2
2
ðx=aÞ þ ðy=bÞ ¼ 1:
ð2Þ
Under τxz as the only nonzero component of the stress acting in the
medium, l = b and C = 3η / 4, where η is represented by the first and
second complete elliptic integrals, for a ≥ b and for a b b, respectively
(Eqs. (5.3) and (5.7), Eshelby, (1957)). The values of a and b are determined by equating πab = WL with a / b = W / L, where W is the fault
width. When L / W is sufficiently large, it is better to use the stress
drop formula for an infinitely long dip-slip fault (Starr, 1928),
Δσ ¼ ð16=3πÞμD=W:
ð3Þ
I use Eq. (3) for L / W N 6 where Δσ calculated for an elliptic crack
approaches asymptotically to that of a dip-slip fault. A circular fault produces 1.44 times larger Δσ than a dip-slip fault for the same D / W.
These formulae are for faults buried sufficiently deep. For Δσ values
of shallow faults close to the Earth's surface, formulae in which W is replaced by 2 W are often used (e.g., Kanamori and Anderson, 1975).
However, in fact, it is a continuous function of d / W, where d is the
depth of the upper edge of the fault (Parsons et al., 1988). Taking into
account the numerical calculations by Parsons et al. (1988), I use the formula for deep (buried) faults when d / W ≥ 0.2, and the formula for
shallow (open) faults when d / W b 0.2, in which W is replaced by
2 W. For a shallow fault, because W is multiplied by 2, the dip-slip
fault formula is applied when L N 6(2 W) = 12 W.
199
2.1. Data sources and fault parameters
I compile studies that obtained slip distributions for Mw ≥ 7 thrusttype earthquakes in subduction zones by inverting long-period seismic
wave, tsunami, or geodetic (GPS or InSAR) data. A rupture zone of an
event is taken to be the area that has more than a few meters slip,
based on which I measure W and L. If the area is not rectangular, I divide
it into subfaults with areas that mimic rectangles (Fig. 1a). Letting
L = ΣLi, I define W as
W ¼ ∑Wi Li =L;
ð4Þ
where Li and Wi are the fault length and width of the i-th subfault, respectively. Fig. 1b shows an example of the division of the rupture
zone into the subfaults for the slip area of the 2011 Tohoku-oki earthquake of Lay et al. (2011).
If only long-period body waves are used to obtain Mo in seismic
studies, other studies that obtained Mo using long-period surface
waves, free oscillations, or gCMT are referred to. I include the fault parameters of the 1700 Cascadia earthquake in the list even though they
were not obtained by inversion because these parameters are
constrained fairly well by the tsunamis (Satake et al., 2003) and the
thermal model (Hyndman and Wang, 1995). When multiple studies
are available for the same event, which often occur for great earthquakes with Mw ≥ 8, I select a solution with the best resolution. I
show the variability in Δσ available from other solutions in Fig. S1 and
listed their values with the data source in Table S1. This indicates that
the selection of other solutions does not seriously affect the results in
this study.
I exclude the April 1, 1968 Mw 7.5 earthquake (Yagi et al., 1998)
southeast of Kyushu (the so-called Hyuga-nada region) from the analyses, because this event belongs neither to the Nankai Trough nor to the
Ryukyu Trench, and it is the only event in this narrow segment. I exclude the 1923 Kanto earthquake in the Sagami Trough, because it
has both the strike-slip and dip-slip components (e.g., Ando, 1971).
The November 12, 1996 S. Peru Mw 7.7 earthquake (Pritchard et al.,
2007) is also excluded, because this event has an extraordinary large
Δσ of 21 MPa. Because S. Peru belongs to the high Δσ segments, as
will be shown later, this exclusion does not change the conclusion in
this study.
Table 1 lists the 53 analyzed events with the estimated fault parameters, data-type, fault-type and data source. Fault-type indicates whether the elliptic or dip-slip formula and the open or buried formula are
used. The parameters of the above 1968 and 1996 events, and three additional events in the Tohoku-oki region in 1793 and 1897 (Aida, 1977)
and in 1992 (Hino et al., 1996; Kawasaki et al., 2001), which have lower
data quality and are not used in the analyses below, are also listed in
Table 1.
2.2. Estimation of μ and depth
In geodetic or tsunami studies, D is generally derived first, and then
Mo is calculated by μDS in each study. However, the assigned μ is sometimes inadequate for the source depth. In this study, I estimate μ at the
average depth of the rupture zone, as stated below, and recalculate
Mo. For the seismic studies, I calculate D from Mo using the estimated
μ and S. From these D and μ, I calculate Δσ using Eq. (1).
The depths of the shallow and deep edges of the rupture zone are estimated from the plate boundary geometry in each region. The source
depth is the average of these depths. The value of μ is calculated at
this depth using the S-wave velocity converted from the P-wave velocity of the forearc model of Mooney et al. (1998), assuming a Poisson's
ratio of 0.25, and the density converted from the density/velocity profile
of Ludwig et al. (1970). The velocity, density and μ as a function of the
depth are listed in Table 2. The depth d of the shallow edge of the rupture zone estimated here is also used to calculate d / W.
200
T. Seno / Tectonophysics 621 (2014) 198–210
Fig. 1. (a) Procedure to determine the fault width W when the rupture zone is not rectangular. Let it be represented by a series of the rectangles with Li and Wi, the i-th fault length and the
width, respectively. W is defined by the arithmetic mean of Wi with a weight of Li/L. (b) Example of the application of (a) to the 2011 Tohoku-oki earthquake for which the slip distribution
was obtained by Lay et al. (2011).
3. Results
3.1. Stress drops over the world subduction zones
I examine Δσ values to see whether a systematic difference exists
between the zones where Mw ≥ 9 earthquakes have occurred and the
zones without such events. I define class 1 earthquakes as Mw ≥ 9
events, class 2 earthquakes as Mw b 9 events in a subduction zone
segment that had at least one Mw ≥ 9 earthquake historically, and
class 3 earthquakes as events in a subduction zone segment that had
no Mw ≥ 9 earthquake. The division of the subduction zones that follows
this classification is shown in Fig. 2. Eight Mw ≥ 9 earthquakes are
known historically: 1833 and 2004 Sumatra, 2011 Tohoku, 1964 Alaska,
1700 Cascadia, 1868 S. Peru, 1877 N. Chile, and 1960 S. Chile earthquakes. These class 1 earthquakes define the segments where class 2
events occur. The 1952 Kamchatka earthquake is excluded from class
1 based on Mw 8.7 derived by Johnson and Satake (1999) using tsunamis. The earlier estimate by Kanamori (1976) of Mw 9 is not reliable
due to malfunctioning of the strainmeter at Pasadena. The following estimates using long-period surface waves (e.g., Okal, 1992) give ~Mw 8.8,
which is more consistent with the estimate by Johnson and Satake
(1999). The 1957 W. Aleutian earthquake is excluded for a similar reason (Johnson et al., 1994). The 1877 N. Chile earthquake is regarded as
class 1 (Mw 9, Schurr et al., 2012), although its magnitude is disputed
(e.g., Comte and Pardo, 1991).
The above choice of Mw ≥ 9 for defining class 1 events may seem an
arbitrary threshold. If the earthquake size is self-similar, it does not have
meaning. What, however, I try to conduct below is to examine whether
there is a difference in Δσ between class 1–2 and class 3 zones, which
may suggest the difference in frictional properties at the thrust between
these classes. It also might be useful for hazard mitigation to conduct attempts to infer which subduction zones can have Mw 9 earthquakes, following the spectacular failure of earlier models.
Three class 1 earthquakes, Alaska, Cascadia, and S. Chile, occur in
segments that do not have any class 2 events in this dataset. Other
class 1 segments include at least two class 2 events. These class 1 events
also include at least one class 2 event within their rupture zone, except
the 2004 Sumatra–Andaman earthquake.
Values for Δσ are plotted for each class in Fig. 3a, with the crosses indicating the average stress drop (Δσ). Δσ values of the class 1 and 2
events are 4.6 and 3.4 MPa, respectively, and are a few times larger
than that of the class 3 events (1.6 MPa). Fig. 3b shows Δσ values of
earthquakes in each subduction zone segment, with the crosses indicating the Δσ values for the class 2 and class 3 events. Because the Δσ
values of the class 2 events are greater than 3.2 MPa, and those of the
class 3 zones are less than 1.7 MPa, except for Kuril–Hokkaido, the results in Fig. 3a also hold for individual subduction zones. Kuril–Hokkaido has a Δσ = 3.2 MPa, which is comparable to those of the class 2
events, although it belongs to class 3. It has recently become known
that prehistoric megathrust earthquakes, much larger than recent historical ones, generated tsunami deposits over coastal areas in southeastern Hokkaido during the Holocene more than ten times (e.g., Nanayama
et al., 2003; Sawai et al., 2009). The high Δσ in this region may indicate
that it in fact belongs to class 2.
Allmann and Shearer (2009) obtained low Δσs in Central America,
which is conformable to the low Δσs in Mexico–Central America in
T. Seno / Tectonophysics 621 (2014) 198–210
201
Table 1
Fault parameters of the subduction zone thrust-type earthquakes.
Region
Event
Date
Sumatra
Sumatra–Andaman
Nias
Mentawai
Bengkulu
Java
Java
Hyuga-nada
Nankai
Tonankai
Shioya-oki
Shioya-oki
Shioya-oki
Miyagi-oki
Miyagi-oki
Miyagi-oki
Miyagi-oki
Tohoku-oki
Tohoku-oki
Java
Hyuga-nada
Nankai
Tohoku-oki
Kuril–Hokkaido
Kamchatka
Aleutian
Alaska
Cascadia
Mexico–C. America
N. Peru
S. Peru
N. Chile
C. Chile
S. Chile
Solomon
μ
(1010 Pa)
Data
type
Fault
type
D
(m)
Δσ
(MPa)
27.5
31
3
26
18
17
17
15.5
21
21
19
22
34
36
45
15
24
17
6.9
6.9
3.2
5.2
4.1
4.1
4.1
3.2
4.1
4.1
4.1
4.1
6.9
6.9
6.9
3.2
5.2
4.1
st
g
gt
gt
s
s
s
t
s
s
s
s
s
s
s
s
s
g
od
oe
oe
oe
oe
oe
be
oe
oe
be
oe
be
be
be
be
oe
oe
be
8.6
4.6
3.8
5.6
.6
.7
2
2.9
1.9
2.4
2.7
2.7
1.2
.9
1.9
.5
13.9
1.2
3.6
3.3
3.2
3.3
0.3
0.4
3.7
1
0.7
4.3
2.3
4.4
4.1
5.4
7.1
0.5
5.3
3
120
120
65
52
13
15
17
14
3.2
3.2
4.1
3.2
t
t
s
s
be
be
be
oe
3.5
3.8
.3
.3
7.0
7.6
0.5
0.3
57
50
110
96
255
100
70
95
180
325
270
609
215
295
240
500
725
1100
200
50
135
53
95
255
110
250
115
105
210
190
165
255
540
850
265
160
27
25
21
30
39
29
33
35
24.5
30
16
20
21.5
19
28
28
36.5
15
15
23
22.5
20
14
5
7.5
30
27.5
30
29
43
23
27.5
22.5
22
18
30
6.9
5.2
4.1
6.9
6.9
6.9
6.9
6.9
5.2
6.9
3.2
4.1
4.1
4.1
6.9
6.9
6.9
3.2
3.2
5.2
5.2
4.1
3.2
3.2
3.2
6.9
6.9
6.9
6.9
6.9
5.2
6.9
5.2
4.1
4.1
6.9
s
s
s
s
s
s
s
s
s
s
s
t
s
t
s
s
g
t
s
s
s
s
s
t
s
s
sg
sg
sg
g
sg
s
g
g
s
s
be
be
be
be
be
oe
oe
be
oe
oe
oe
oe
oe
oe
oe
oe
oe
od
oe
oe
oe
be
oe
oe
oe
oe
oe
be
oe
be
be
oe
oe
oe
oe
be
.3
.3
2.8
.2
2.8
2.5
2.9
1.8
3.9
3.2
5.7
3.2
1
5.3
.7
2.2
11.5
14
1.5
1.1
3.2
1.2
.5
3
1.4
1.2
2.8
3.3
4.6
1.2
3.8
.7
5
17
2
1.4
0.8
0.8
6.1
0.4
5.2
2.8
4.3
4.4
3.6
2.3
2
0.7
0.5
1.8
0.5
0.9
4.4
5.4
.7
1.7
2.6
2.2
.3
2.3
1.2
1.2
3.4
21
3.9
3.4
6.8
0.5
1.9
5
1
2.4
Mo
(1020 Nm)
Mw
W
L
(km)
Depth
12/26/2004
03/28/2005
10/25/2010
09/12/2007
07/17/2006
06/02/1994
04/01/1968*
12/20/1946
12/07/1944
05/23/1938
11/05/1938
11/05/1938
11/02/1936
08/16/2005
06/12/1978
01/18/1981
03/11/2011
03/09/2011
1000
98
6.3
70
5.6
3.5
2.5
31.5
24
3.1
6.9
4.5
2.2
0.54
3.1
0.56
400
0.75
9.3
8.6
7.8
8.5
7.8
7.6
7.5
8.3
8.2
7.6
7.8
7.7
7.5
7.1
7.6
7.1
9
7.2
140
105
40
120
110
100
48
105
140
50
70
55
56
33
80
54
170
53
1200
295
130
200
215
130
64
320
220
65
90
75
46
26
30
60
325
29
Miyagi-oki
Miyagi-oki
Iwate-oki
Iwate-oki
08/05/1897*
02/17/1793*
06/12/1968
07/18/1992*
4.04
4.44
0.4
0.3
7.7
7.7
7
6.9
30
30
57
52
Iwate-oki
Iwate-oki
Sanriku–Haruka-oki
Aomori-oki
Tokachi-oki
Tokachi-oki
Tokachi-oki
Nemuro-oki
S. Kuril
S. Kuril
C. Kuril
Kamchatka
Cape Kronotsky
Andreanof Is.
Andreanof Is.
Rat Island
Alaska
Cascadia
Jalisco
Tecoman
Michoacan
Playa Azul
Petatlan
Nicaraguan
N. Peru
N. Peru
Pisco
S. Peru
S. Peru
Tocopilla
Antofagasta
C. Chile
Maule
S. Chile
Solomon
Solomon
03/20/1960
11/01/1989
12/28/1994
03/09/1931
05/16/1968
03/04/1952
09/26/2003
06/17/1973
08/12/1969
10/13/1963
11/15/2006
11/04/1952
12/05/1997
03/09/1957
05/07/1986
02/04/1965
03/28/1964
01/26/1700
10/14/1995
01/22/2003
09/19/1985
10/25/1981
03/09/1979
09/02/1992
02/21/1996
10/03/1974
08/15/2007
11/12/1996*
06/23/2001
11/14/2007
07/30/1995
03/03/1985
02/27/2010
05/22/1960
04/01/2007
07/14/1971
1.0
0.4
4.4
0.9
35
23
17
7
22
75
50
156
7.2
88
13
140
1069
348
8.3
2.3
20
1.35
1.5
10
2
15
17.8
4.8
63
7
18
15
197
761
18.7
12
7.3
7
7.7
7.2
8.3
8.2
8.1
7.8
8.2
8.5
8.4
8.7
7.8
8.6
8
8.7
9.3
9
7.9
7.5
8.1
7.4
7.4
7.9
7.5
8.1
8.1
7.7
8.5
7.8
8.1
8.1
8.8
9.2
8.1
8
73
50
35
69
70
130
120
60
60
105
100
197
84
140
110
180
185
70
85
80
90
53
95
40
40
75
80
20
95
45
55
123
140
130
85
80
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Kikuchi and Fukao (1987)
Suito and Freymueller (2009)
Satake et al. (2003)
Mendoza and Hartzell (1999)
Yagi et al. (2004)
Mendoza and Hartzell (1989)
Mendoza (1993)
Mendoza (1995)
Satake (1994)
Ihmle et al. (1998)
Hartzell and Langer (1993)
Pritchard and Fielding (2008)
Pritchard et al. (2007)
Pritchard et al. (2007)
Schurr et al. (2012)
Pritchard et al. (2006)
Mendoza et al. (1994)
Pollitz et al. (2011)
Barrientos and Ward (1990)
Furlong et al. (2009)
Kikuchi and Fukao (1987)
Regions and earthquakes are put in the order of a clock-wise sense geographically, except Sumatra and Java, where they are from west to east. Earthquakes with * are not used in the analyses. s, g, and t in Data type denotes seismic, geodetic and tsunami data, respectively, used in the inversion for the slip distribution. o or b, and e or d in Fault type indicate whether open or
buried, and elliptic or dip-slip formulae are used.
this study. However, they did not recognize a significant difference in
Δσ for other subduction zones. It should be noted that they obtained dynamic Δσs, assuming a circular fault with a constant rupture velocity.
The method cannot be applied to large earthquakes, and their absolute
values of Δσ should not be directly compared to those of the present
study.
The Δσ s in Fig. 3 for class 2 events are dispersed and contain some
low values. Fig. 3b shows such low Δσs come from the Tohoku-oki
region. In order to see what area they are distributed, I show the spatial
distribution of the Δσ values and rupture zones for earthquakes in the
Tohoku-oki region in Fig. 4. In this figure, those of the 1793 and 1897
historical earthquakes and the 1992 earthquake are also plotted to supplement the data. These earthquakes are not used in the statistics of Δσ,
because they have lower quality. Δσ values of the earthquakes in the
rupture zone of the 2011 Tohoku-oki earthquake are evidently high.
In contrast, those located north of the rupture zone contain events
202
T. Seno / Tectonophysics 621 (2014) 198–210
be proved completely, nor it is proved that Mw ≥ 9 earthquakes do
not occur in class 3 subduction zones.
Table 2
Velocity and rigidity structure.
Vp
(km/s)
Vs
(km/s)
ρ
(kg/m3)
μ
(1010 Pa)
Thickness
(km)
Top.depth
(km)
2.3
4
6
6.6
7.2
8
1.33
2.31
3.46
3.81
4.16
4.62
2.03
2.27
2.7
2.79
3
3.25
.36
1.21
3.23
4.05
5.19
6.94
1.1
1
14.9
5.2
4.7
–
0
1.1
2.1
17
22.2
26.9
4. Scaling relations
To date, in the scaling relations between D, W, L, and Mo, Δσ has
been treated as a scale-independent constant (e.g., Hanks, 1977;
Kanamori and Anderson, 1975; Romanowicz, 1992). I have observed,
however, that there is a difference in Δσ between the class 1–2 and
class 3 events. To understand why large Δσ can arise for Mw ≥ 9 earthquakes, I explore the effects of the variable Δσ on the scaling relations. I
first present the relations between D, W, L and Mo, noting the difference
between the classes, and then derive the scaling relations between D
and S and between L and Mo that take into account the variable Δσ.
Vp is from the forearc model of Mooney et al. (1998). Vs is obtained from assuming the
Piosson's ratio of 0.25. Density ρ is read from the velocity in Ludwig et al. (1970).
with low Δσ values. One ad-hoc explanation to this behavior might be
that the region north of the 2011 Tohoku-oki earthquake constitutes another segment.
4.1. Relations among the parameters
I plot W and D versus L in Fig. 5a and b, respectively, with different
symbols for the class 1, 2, and 3 events. W values of the class 1 events
are generally greater than 100 km; the small value of W (~60 km) for
the 1700 Cascadia earthquake is anomalous, and this narrow
seismogenic zone results from the subduction of the young Juan de
Fuca plate (Hyndman and Wang, 1995). W of the class 1 and 3 events
reaches the maximum W0 of 150–200 km at L ~ 600 km. The saturation
of W might be a phenomenon similar to the limit of the depth extent of
the continental strike-slip earthquakes (e.g., Romanowicz, 1992; Scholz,
1982; see Oleskevich et al., 1999 for the down-dip limit of subduction
zone thrust-type earthquakes). Thus Mw b 9 for the class 3 events
does not result from low values of W. The solid line in Fig. 5a represents
3.2. A criterion to differentiate the zones where Mw ≥ 9 earthquakes occur
The above results suggest a possibility of utilizing Δσ in a subduction zone segment to differentiate the zones where Mw ≥ 9 earthquakes
possibly occur. Based on Fig. 3, I propose a hypothesis that if Δσ in a
segment is greater than 3 MPa, Mw ≥ 9 earthquakes could occur in
this zone, and if Δσ is less than 2 MPa, they would not. If this hypothesis is correct, in Kuril–Hokkaido a Mw ≥ 9 earthquake is expected to
occur in the future, because its Δσ is greater than 3 MPa. Because the
seismicity data are historical and limited, the above hypothesis cannot
1964
60 N
Kamchatka
Ryukyu
Mariana
Mexico
C. America
So
ma
lom
Columbia
20
1868
S. Peru
N. Chile
Kermad
tu
nua
Va
Sunda
ec Tonga
tra
on
1833
L. Antilles
2004
u
an S
Latitude
2011
ne
ippi
Andam
0
1700
Bonin
Phil
20
Ku
adia
40
Casc
Aleutian
ril
Tohoku
Nankai
a
ask
Al
1877
C. Chile
N. Peru
i
ng
a
ur
40
S. Chile
ik
H
1960
tia
Sco
60 S
80
100
120
140
160
180
200
220
240
260
280
300
320
340
Longitude
M
9 earthquakes
19th
Class 1 + Class 2
20th
Class 3
21th
Data not available
Fig. 2. Subduction zones are classified into class 1 - 2 where Mw ≥ 9 earthquakes occurred and class 3 where earthquakes occurred in a subduction zone segment in which no Mw ≥ 9
earthquake has occurred. Eight historical class 1 earthquakes are marked by large circles: 1833 and 2004 Sumatra, 2011 Tohoku, 1964 Alaska, 1700 Cascadia, 1868 S. Peru, 1877 N. Chile,
and 1960 S. Chile earthquakes. These earthquakes define class 1 and 2 segments. The class 1–2 and the class 3 segments are shown in red and green, respectively. Subduction zones with
no fault parameter data are indicated in black.
T. Seno / Tectonophysics 621 (2014) 198–210
a
203
43˚
8
Hokkaido
3 MPa < Δσ
Δσ (MPa)
6
2 MPa < Δσ < 3 MPa
42˚
Δσ < 1 MPa
4.6 MPa
4
3.4 MPa
1968
5.2
41˚
2
40˚
1
2
3
0.8 0.8
1960
1989
Class
1992
Tohoku
b
1968
1793
3.9
N. Chile
4
Sumatra
38˚
4.4
Nankai
7.0
2011
1938
3.11
5.3
Fukushima
Solomon
1897
1936
S. Chile
Mexico
37˚
2.3
1938
N. Peru
4.3
1938
Java
0
0.5
1981
5.4
4.0 2005
Aleutian
2
3.0
1978
7.1
Cascadia
KurilHokkaido Alaska
S. Peru
7.6
2011
Miyagi
6
0.3
0.5
39˚
8
Δσ (MPa )
6.1
Japan T
r
0
0.4
1931
ench
1994
1.6 MPa
Tohoku Kamchatka
C. Chile
Kanto
36˚
km
Fig. 3. (a) Stress drops (Δσ) of the class 1 (red circles), class 2 (blue circles), and class 3
(green circles) earthquakes. Their average stress drop (Δσ) values are 4.6 and 3.4, and
1.6 MPa, respectively, and are shown by the black crosses. Those of the class 1 and 2 earthquakes are a few times greater than those of the class 3 earthquakes. (b) Stress drops (Δσ)
of the class 1, class 2, and class 3 earthquakes and their averages (Δσ) are shown by the
red, blue, and green solid circles and crosses, respectively, in each subduction zone segment. Δσ of the class 2 events are greater than 3 MPa, and Δσ of the class 3 events are
less than 2 MPa. Kuril–Hokkaido is an exception, which has a Δσ of 3.2 MPa, that is comparable to Δσ values of the class 2 events.
the relation between W and L that is expected from the scaling relation
between L and Mo, which is derived later.
In Fig. 5b, although it is not clear, as a whole, whether D follows
a linear trend (L-models) or saturates at a certain value around W0
(W-models), the distinction in D among the classes is clear. The D values
of the class 1 events are significantly greater than those of other classes,
which are below ~5 m. It is also noted that the D values for the class 2
events are a few times greater than those of the class 3 events. The D
values in each class show a trend increasing and then gradually saturating along with L, which is similar to that observed for the continental
interplate earthquakes (Fig. 1 of Scholz, 1994). I will show that these
features can be explained by the scaling relation between D and L,
that is derived later (the solid lines).
I plot LogL versus LogMo in Fig. 6 with the slopes of 1/3, 1/2 and 1,
which correspond to D ∝ W –L, D ∝ L and L ∝ S (L-models), D =
constant and L ∝ S (W-models), respectively (e.g., Romanowicz,
1992). Although there is a large scatter, LogL versus LogMo of the class
2 and 3 events are between the slopes of 1/2 and 1/3. LogL of the class
2 events are generally sifted downward from those of the class 3 events,
which indicates that the former have greater Δσ than the latter. Because
of the scarcity of the class 1 earthquakes, the plot is not able to fit any
slope to these events. The data in Fig. 6, having an extent of scatter similar to the previous studies of dip-slip earthquakes (Blaser et al., 2010;
Henry and Das, 2001; Murotani et al., 2013), make the scaling relation
between LogL and LogMo not useful for predicting Mo based on L. I
0
100
143˚
144˚
200
35˚
139˚
140˚
141˚
142˚
145˚
Fig. 4. Rupture zones and Δσ values of the Mw ≥ 7 interplate earthquakes in the Tohoku-oki
region. The value of Δσ is shown in the brown rectangle. The rupture zones are indicated in
pink (Δσ ≥ 3 MPa), yellow (3 MPa N Δσ ≥ 2 MPa) and green (2 MPa N Δσ). The broken line
shows the rupture zone of the 2011 Tohoku-oki 2011 earthquake (Lay et al., 2011). The
1793 and 1897 and the 1992 earthquakes, whose fault parameter data have lower quality
and are not used in the statistical analyses, are also shown. Most earthquakes in the rupture
zone of the 2011 event have Δσ ≥ 3 MPa, except the 1981 event (Δσ = 0.5 MPa), and the
earthquakes with small Δσ values (2 MPa N Δσ) are distributed north of it.
will show below that the scatter is caused by the effects of Δσ and reduced by normalization of Mo using Δσ.
4.2. Stress drop and asperity area
In this subsection, I introduce the relation between Δσ and the fraction of the asperity areas in the fault area, to derive the scaling relations
between D and S and between L and Mo, which take into account the
variation in Δσ. I divide the fault plane into a number of asperities and
surrounding barriers. Because Δσ has been obtained from D averaged
over the fault plane that contains asperities, Δσs over individual asperities are different from Δσ. I relate Δσs over asperities to Δσ in order to
obtain the scaling relations.
Asperities are defined as the portions of the fault area that have an
unstable frictional property and adhere during the interseismic periods.
Barriers are defined as portions with a stable sliding frictional property. I
put a subscript A on the variables that are related to an asperity. The
total shear force over the fault plane is partitioned into those over asperities and barriers as
τS ¼
X
τAi SAi þ
X
τBj SBj ;
ð5Þ
where τ is the average shear stress over S, and τΑi is the shear stress over
204
T. Seno / Tectonophysics 621 (2014) 198–210
a
Slope of 1
3
150
L= cW
Log (L/km)
W (km)
200
3
Slope of 1/3
2
100
Slope of 1/2
50
1
0
4
8
0
12
2
Fig. 6. (a) Plot of LogL versus LogMo. The slopes of 1/3, 1/2 and 1 are shown by the solid,
chain and broken lines, respectively. These slopes correspond to earthquakes with
W ~ L, L-models, and W-models, respectively (Romanowicz, 1992; Scholz, 1982). The
data of the class 2 and 3 events are between the slopes of 1/3 and 1/2, although they are
diffuse. Those of the class 2 events are shifted downward than those of the class 3 events,
indicating that they have higher Δσ.
b
D (m)
15
the asperities, thus yielding
10
Δσ ¼ Δσ A SA =S:
D= cL
0.67
5
0.67
D= 0.3cL
0
4
8
12
L (100 km)
Fig. 5. (a) Plot of W versus L. In this figure and the following Figs. 6 and 7, the data of the
class 1, class 2, and class 3 events in Table 1 are plotted by the red, blue and green circles,
respectively. W saturates approximately 150–200 km for the class 1 and 3 events. A solid
line shows L = cW3, which is derived from Eq. (17) (see the text). (b) Plot of D versus L. D
for the class 3 events are smaller than those of the class 1–2 events. The relation between D
and L (Eq. 18) is shown by the solid line for each of class 1–2 and class 3 events (see text).
the i-th asperity with an area SΑi and τBj the shear stress over the j-th
barrier with an area SBj. Asperity areas occupy only a fraction of the
fault area (e.g., Murotani et al., 2008; Ruff and Kanamori, 1983b). Because propagation of a rupture that follows breakage of one asperity is
prohibited by surrounding barriers (Kato and Hirasawa, 1999; Tse and
Rice, 1986), Seno (2003) regarded barriers within the rupture zone as
having a nearly zero friction, with the pore fluid pressure elevated
close to the lithostatic pressure (denoted by “invasion of barriers”).
Kanamori and Allen (1986) and Ruff and Kanamori (1983a) similarly
assumed barriers as having ~0 strength, without noting the pore fluid
pressure. Assuming τBj ~ 0 in Eq. (5), I obtain
τS ¼
X
τAi SAi :
ð6Þ
For simplicity, taking τΑi as uniform (=τΑ) over the asperities in the
fault, Eq. (6) becomes
τS ¼ τA SA ;
4
20
Log (Mo/10 Nm)
L (100 km)
ð7Þ
where SΑ = Σ SΑi, that is, the total area of the asperities. When a rupture
occurs over the fault, Δσ over S is sustained by the stress drop ΔσΑ over
ð8Þ
Madariaga (1979) obtained an expression similar to Eq. (8) (Eq. 15
of Madariaga, 1979). Because ΔσΑ is 5–10% of σn, where σn is the normal
stress of the fault (Beeler, 2001; Johnson et al., 1973; Kato, 2012;
Ohnaka et al., 1997) and σn is nearly constant for the depths of the
earthquakes treated in this study (i.e., d = 20–30 km), I neglect the
depth-dependence of ΔσΑ. Thus, Δσ becomes a parameter that represents SΑ/S, a fraction of the asperity areas in the total fault area. In the
following subsection, based on Eq. (8), I derive the scaling relation between D and S that takes into account the variation in Δσ.
4.3. Scaling relation between D and S
As I state in the previous subsection, to the first order, I regard ΔσΑi
as being constant, thus
LAi ∝DAi :
ð9Þ
Because SΑi ~ L2Αi, Eq. (9) gives SΑi ∝ D2Αi. Putting this into Eq. (8), I
obtain
X
ΔσS ∝
SAi
∝
X
2
DAi :
ð10Þ
On the other hand, the sum of the moments of the asperities equals
to the total moment,
DS ¼
X
DAi SAi ∝
X
3
DAi :
ð11Þ
Let DΑ represent any DΑi when there are asperities with similar sizes,
or that of the asperity that has a dominant size. Eqs. (10) and (11) give
ΔσS ∝ DA
2
3
and DS ∝ DA :
ð12Þ
(For example, this approximation to Eqs. (10) and (11) gives only a
4% error in the case of two asperities with the dominant size being twice
T. Seno / Tectonophysics 621 (2014) 198–210
Until W saturates at the peak value of Wo, W ~ L and thus S ~ L2, then
that of the minor one). Removing DΑ from Eq. (12), I obtain
2=3 2=3
ΔσS ∝ D
S
or
3
2
Δσ S ∝ D :
ð13Þ
This scaling relation between D and S takes into account the variation in Δσ. Note that this relation is not the definition of Δσ, because
Δσ is defined by Eq. (1). In Fig. 7a, Δσ3S is plotted versus D using the
data listed in Table 1, which is fit fairly well by the curve Δσ3S = cD2,
where c is an arbitrary constant.
4.4. Scaling relation between L and Mo
I derive the scaling relation between L and Mo, introducing Δσ as a
normalizing parameter. First, using Eq. (13), I obtain the relation between Mo, Δσ, and S as
1=2
3
1:5 1:5
Mo ¼ μDS ∝ Δσ S
S ¼ Δσ S :
ð14Þ
12
Δσ3 S (106 MPa3 km2)
Mo ∝ Δσ
1:5 3
L
1:5
or LogL ∝ 0:33Log Mo =Δσ
:
ð15Þ
After W saturates at Wo, S ∝ L, then
Mo ∝ Δσ
1:5 1:5
L
or
1:5
:
LogL ∝ 0:67Log Mo =Δσ
ð16Þ
In Fig. 7b, I plot LogL versus Log(Mo/Δσ1.5) using the data in Table 1,
with the slopes of 0.33 and 0.67. The systematic difference in the slope
between the class 1 events and the class 2–3 events is not evident,
which indicates that the above scaling between L and S might not
hold. However, the general linear trend suggests that the rupture of subduction zone earthquakes, from Mw ≥ 9 through Mw ~ 7, can be understood on the basis of the same scale-invariant physics, if Mo is
normalized by Δσ1.5. As an application of this scaling, I try to predict
Mo from L for an earthquake that may have the greatest lateral rupture
extent in the Nankai Trough in the next subsection.
4.5. Magnitude expected for the largest earthquake in the Nankai trough
a
8
3
Δσ S = cD
4
0
0
5
10
2
15
20
D (m)
b
Slope of 0.50
3
Log (L/km)
205
Slope of 0.67
2
Slope of 0.33
1
-2
-1
0
1
2
3
Along the Nankai Trough, where the Philippine Sea plate is
subducting in the WNW direction beneath the Eurasian plate (Seno
et al., 1993), great earthquakes have occurred historically (e.g., Ando,
1975). The rupture zones of the recent 1944 Tonankai (M 7.9) and
1946 Nankai (M 8.0) earthquakes are shown in Fig. 8, along with
those of the 1854 Ansei–Tokai (M 8.4) earthquake in the Suruga Trough,
which is contiguous to the eastern terminus of the Nankai Trough, and
the 1923 Kanto (M 7.9) earthquake in the Sagami Trough further to
the east. Because the rupture zone of the Ansei–Tokai earthquake did
not break at the time of the Tonankai and Nankai earthquakes, a future
event that might break this segment was anticipated (Ishibashi, 1981).
The Large-Scale Earthquake Countermeasures Act was enacted in 1978
by the Japanese parliament to prepare for this postulated earthquake
(the so-called Tokai earthquake). Since then, more than 30 years have
passed, and recently, the Central Disaster Management Council,
Cabinet Office of the Japanese Government (2011) issued an advisory
that the earthquake, in the worst case, may involve a simultaneous occurrence of the Tokai, Tonankai and Nankai earthquakes, with a maximum magnitude of Mw 9.1 and damage amounting to 0.2 billion
dollars. The expectation of such a large magnitude may be influenced
by the unexpected size of the 2011 Mw 9 Tohoku-oki earthquake.
Conversely, Seno (2012) showed that the Ansei–Tokai and Tonankai
rupture zones that generated seismic waves are complementary to each
other, and thus the unbroken Tokai rupture zone does not necessarily
mean that a great earthquake is anticipated there in the near future.
Even in this case, however, it is not impossible, over a long geologic
time, for these earthquakes to occur simultaneously. I therefore estimate the magnitude of such an earthquake using the scaling relation obtained above. I extend the western edge of this event further to the west
in the Hyuga-nada region (Fig. 8), following the Central Disaster Management Council. I also extend the rupture zone to the Suruga Trough.
The total length L amounts to 665 km (the blue thick broken line in
Fig. 8).
The data in Fig. 7b are best fitted by the least-squares line giving
Log(Mo/Δσ1.5) as a linear function of LogL. The best fit (dotted line) is
given by
1:5
þ 1:85;
LogL ¼ 0:5Log Mo =Δσ
ð17Þ
Log (Mo/Δσ1.5 /10 Nm/MPa1.5)
Fig. 7. (a) Plot of Δσ3S versus D. The solid line represents Δσ3S = cD2 derived from
Eq. (13), where c is an arbitrary constant. (b) Plot of LogL versus Log(Mo / Δσ1.5). The
slopes of 0.33 and 0.67 that are expected from the scaling relations S ~ L2 and S ∝ L, respectively, are shown by the solid lines (see text). The broken line shows the least-squares
fitting that regards Log(Mo / Δσ1.5) as a linear function of LogL. This line has the slope of
0.5 with the intercept of 1.85.
with s.d. of 0.41. The slope of 0.5 between 0.33 and 0.67 implies that
S ~ L2 or S ∝ L in Eq. (15) or Eq. (16) does not hold, and gives L ∝
W3, that is, S ∝ L1.33. This curve between L and W is plotted in Fig. 5a,
which roughly fits the data except for the largest class 1 earthquakes, although other types of curves may fit the data equally well. I therefore
use Eq. (17) to estimate Mo and obtain Mw = 8.6–8.4, using the Δσ
206
T. Seno / Tectonophysics 621 (2014) 198–210
Eurasian plate
M7.9
n Tre
1923 Kanto
1853 Ansei-Tokai
nch
Honshu
36°N
Izu
1944 Toankai
M7.9
Suruga Trough
M8.0
4 cm/yr
Kyushu
1.0
Na
Hyuga-nada
gh
rou
iT
nka
32°
3 cm/yr
0.7 MPa
rench
1946 Nankai
Trough
Philippine Sea plate
0
100 km
Izu-Bonin T
34°
Sagami
Japa
M8.4
5 cm/yr
1946
Nankai
132° E
134°
136°
138°
140°
142°
Fig. 8. Rupture zones of the historical great earthquakes in the Nankai–Suruga–Sagami Troughs: the rupture zones are from Ando (1971) for the 1923 Kanto earthquake (purple), Ando
(1975) for the 1946 Nankai earthquake (green), and Ishibashi (1981) for the 1944 Tonankai (blue) and 1854 Ansei–Tokai (yellow) earthquakes. Δσs of the Tonankai and Nankai earthquakes are also shown. Their magnitudes are from the Scientific Chronological Table (Tokyo Astronomical Observatory, 2012). The gray arrows show the convergence velocities of the
Philippine Sea plate beneath the Eurasian plate (Seno et al., 1993). The length L of an earthquake that is hypothesized to rupture the entire Nankai–Suruga Trough is shown by the
blue thick broken line, which reaches 665 km.
values of 1–0.7 MPa in the Nankai Trough (Table 1). Adding the value of
s.d. to the Log(Mo/Δσ1.5) produces Mw = 8.9. The Δσ value less than
2 MPa in the Nankai Trough predicts that Mw ≥ 9 earthquakes would
not occur there, which is consistent with the obtained Mw. The Mw 9.1
assigned by the Central Disaster Management Council for the earthquake that ruptures all along the Nankai–Suruga Trough seems to be
overestimated.
4.6. Scaling relation between D and L
The scaling relation (13) between D and S gives the scaling relation
between D and L. For L b ~600 m (Fig. 5a), using L ∝ W3, I get
3
3 1:33
Δσ LW ∝ Δσ L
2
∝D
or
D ∝ Δσ
1:5 0:67
L
;
ð18Þ
and for L N ~600 km, W saturates at W0 and L ∝ S. Thus, I obtain
3
3
2
Δσ S ∝ Δσ L ∝ D
or D ∝ Δσ
1:5 0:5
L
:
ð19Þ
In the plot of D versus L in Fig. 5b, the curves of Eq. (18) are plotted
for the class 1–2 and class 3 events, calibrating their relative amplitude
using the ratio of Δσ values of these classes (Fig. 3a). The trend that D
increases as Lx, with x ~ 0.5, is also seen in the data in Fig. 1 of Scholz
(1994) for the continental interplate earthquakes. This suggests that
the continental interplate earthquakes may contain asperities and
obey the scaling relation (13), not those of W- or L-models.
5. Discussion
Seismic coupling has been investigated by many researchers (e.g.,
Conrad et al., 2004; Heuret et al., 2011; Jarrard, 1986; Kanamori, 1977;
Lay and Kanamori, 1981; Pacheco et al., 1993; Peterson and Seno,
1984; Ruff and Kanamori, 1980, 1983a; Scholz and Campos, 1995,
2012; Uyeda and Kanamori, 1979). Ruff and Kanamori (1980, 1983a)
proposed that both the age and convergence rate of the subducting
plate are the controlling parameters of the maximum magnitude of
subduction zone earthquakes. As mentioned before, this model was
shown not to work well by Stein and Okal (2007, 2011) for the 2004 Sumatra–Andaman and 2011 Tohoku-oki earthquakes. A proposal has
been made that Mw ≥ 9 earthquakes can occur in any subduction
zone (McCaffrey, 2007, 2008). Meanwhile, various ideas on the relationship between the seismic coupling and the plate tectonic parameters
have been proposed (e.g., Conrad et al., 2004; Gutscher and
Westbrook, 2009; Heuret et al., 2011; Scholz and Campos, 1995,
2012). However, it does not seem yet easy to entangle their relationship.
The results in this study suggest that the Δσ values of the thrusttype earthquakes could be used to differentiate the zones where
Mw ≥ 9 earthquakes possibly occur from those where they are not
anticipated. This finding conforms more to the idea that there is a variation in the seismic coupling (Uyeda and Kanamori, 1979) than the
McCaffrey's (2007) idea. In this section, I try to explain how Δσ affects
Mw and also reflects the hydrological state of the thrust zone.
5.1. Stress drop, asperity size and rupture growth
Provided that ΔσA does not vary much over the depths of subduction
zone earthquakes, Δσ represents SA/S, a fraction of the asperity areas in
the total fault area. Fig. 9 illustrates schematically the difference in SA/S
and the asperity size for the class 1, 2, and 3 events. As a typical example,
the ratio of S of the class 1 and class 2 events is taken to be 4:1 (see
Fig. 5a), and that of the larger and smaller class 3 events is the same.
SA/S for the class 1–2 events is taken to be 0.3, and that of the class 3
events is 0.1 based on the difference in Δσ in Fig. 3a. The number of asperities in the rupture zone is taken to be 6.
The difference in S between the class 1 and class 2 events, both with
similar Δσ, results in the difference in D through Eq. (13) and the asperity size through Eq. (8). The reason that the class 2 events do not grow
up to class 1 events is not obvious. Because they have Δσ values similar
to those of the class 1 events, Heaton's (1990) idea that higher Δσ drives
a rupture to a long distance is not applicable in a straightforward manner. An ad-hoc explanation is to assume that, for the class 2 events, barriers are effective; they might be structural boundaries (Das and Aki,
T. Seno / Tectonophysics 621 (2014) 198–210
where λAi and σnAi are the pore fluid pressure ratio and the normal
stress at the i-th asperity, respectively, and λBj and σnBj are those at
the j-th barrier, respectively. Assuming that barriers are with λBj ~ 1 (invasion of barriers, Seno, 2003), the second term in Eq. (21) vanishes and
the equation is reduced to
b
a
207
Class 1
Class 2
ð1−λÞσ n S ¼
X
ð1−λAi Þσ nAi SAi :
ð22Þ
If (1 − λAi)σnAi does not vary much among asperities and is denoted
by (1 − λA)σnA, Eq. (22) becomes
ð1−λÞσ n S ¼ ð1−λA Þσ nA SA :
ð23Þ
Because σn ~ σnA, this gives
1−λ ¼ ð1−λA ÞSA =S;
Class 3
Class 3
Fig. 9. Schematic illustration that shows the asperity size distribution for (a) the class 2
event, (b) the class 1 event, (c) the smaller class 3 event, and (c) the larger class 3
event. The ratio of S between the class 1 and 2 events is taken to be 4:1, and that between
the larger and smaller class 3 events is the same. SA/S is arbitrarily assigned to be 0.3 for the
class 1 and 2 events, which gives 0.1 for that of the class 3 events through Eq. (8). The area
that includes these asperities is surrounded by un-invaded barriers, which prevent further
rupture propagation (see Seno, 2003).
1977) or un-invaded barriers (Seno, 2003; the shaded areas in light
brown surrounding the rupture zones in Fig. 9). If the asperities have
a fractal structure (Frankel, 1991; Seno, 2003), the size of the largest asperity in the fault zone would increase as the fault area increases, and
may eventually grow up to the size of asperities of the class 1 events.
On the other hand, the class 3 events have smaller SA for the same S
than the class 1 and 2 events (Fig. 9). For a fixed area of the plate interface, the size of asperities contained in this area is smaller for the class 3
events. This geometry results in the smaller DΑ, and the dynamic triggering for the rupture of nearby larger asperities (Ide and Aochi,
2005) might not be favored. This may explain the reason that the class
3 events do not become as large as the class 1 events.
5.2. Pore fluid pressure ratio and stress drop
The value of Δσ is related to the pore fluid pressure ratio λ (Seno,
2009). Considering λ and σn as the average values over the fault
plane, the shear strength τ over the fault plane is
ð24Þ
with the right side essentially identical to that of Eq. (8) if λA is nearly
constant at the depths of the earthquakes in this study.
Although values of λ have been estimated in several subduction
zones (e.g., Davis et al., 1983), reliable estimation, which is critical to
see the variation of SA/S, is scarce (see Seno, 2009; Wang and He,
1999). Seno (2009) proposed a method to determine λ based on the
force balance between τ along the thrust and the differential stress at
the rear of the forearc and estimated λ using the topography and crustal
structures for the Nankai, Miyagi, S. Vancouver Is., Washington, N. Peru,
N. Chile, and S. Chile subduction zones, whose value varies from 0.98
through 0.90. If λ = 0.95 and SA/S = 0.3, λA = 0.17 from Eq. (24).
He showed that Δσ is in fact correlated with 1 − λ in these zones.
Because Δσ values have been newly estimated in this study, I replot
Δσ versus 1 − λ in Fig. 10. For Cascadia, because there are two estimates
of λ in S. Vancouver Is. and Washington, I calculate the Δσ values for
these zones (6.3 and 4.7 MPa, respectively) using the Mo of the 1700
event (Satake et al., 2003) and W = 60 and 80 km, respectively
(Hyndman and Wang, 1995). Fig. 10 shows that a correlation exists between Δσ and 1 − λ (correlation coefficient = 0.915), as is expected,
although the deviations of N. Peru and N. Chile are large. Because the
values of λ can be estimated independently of Δσ, 1 − λ can hopefully
be used as a proxy for Δσ where no earthquake data are available. Unfortunately at present, however, the number of subduction zones
where reliable values of λ have been estimated is limited.
S. Vancouver Is.
6
N. Chile
Washington
Δσ (MPa)
d
c
S. Chile
4
Miyagi
2
τ ¼ τ0 þ μ ðσ n −Pw Þ ¼ τ0 þ μ ð1−λÞσ n ;
where τ0 is the cohesive strength and μ is the coefficient of static friction. Noting that τ0 is negligible for the depths of the subduction zone
thrusts (Byerlee, 1978) and the shear strength is almost equal to the
shear stress, provided that Δσ is only a small fraction of τ as noted previously, I obtain from Eqs. (5) and (20)
X
X
ð1−λÞσ n S ¼
ð1−λAi Þσ nAi SAi þ
1−λBj σ nBj SBj ;
N. Peru
ð20Þ
ð21Þ
Nankai
0
0
0.04
0.08
0.12
1−λ
Fig. 10. Plot of Δσ versus 1 − λ. Values of λ are from Seno (2009). Δσ values are from this
paper, except Cascadia, for which Δσs at S. Vancouver Island and Washington are calculated using W = 60 and 80 km (Hyndman and Wang, 1995), respectively, and the Mo of the
1700 earthquake by Satake et al. (2003).
208
T. Seno / Tectonophysics 621 (2014) 198–210
5.3. Controlling factors of SA/S
As shown above, the values of 1 − λ, Δσ, and thus SA/S represent one
aspect of the seismic coupling. The value of λ represents the extent of
the pore fluid pressure in the fault zone, which would be governed by
the hydrological interaction between the fault gauge and the surrounding materials (e.g., Neuzil, 1995) over a long geologic time. Note that its
temporal change associated with earthquakes (e.g., Sibson, 1992; Sleep
and Blanpied, 1992) is only the order of ~10% of the total amount (Seno,
2009). The source of fluids is dehydration from the metamorphosed
crust and mantle of the subducting slab because the dewatering from
the sediments and clay minerals such as smectite in the oceanic crust
is completed at shallow depths (Fyfe et al., 1978; Moore and Saffer,
2001). The supply of the fluids thus mainly occurs beneath and deeper
than the seismogenic thrust (Hacker et al., 2003; Kirby et al., 1996;
Peacock and Wang, 1999; Yamasaki and Seno, 2003). The dehydration
rate would be controlled not only by the subduction rate or the thermal
structure, that is, the age of the slab (e.g., Anderson et al., 1976), but also
by the extent of the hydration of the slab caused by various mechanisms, such as the bending at the trench (e.g., Faccenda et al., 2009),
the thermal cracking of the oceanic plate (Korenaga, 2007), and the
metamorphism by the H2O–CO2 vapors released from magmas in upwelling plume heads (Kirby et al., 1996; Seno and Yamanaka, 1996).
The escape of the fluids from the fault zone to the forearc wedge of
the upper plate, the trench, and the deep mantle is also controlled by
various factors. It is affected by the tectonic stresses of the forearc
through the ease of hydrofracturing (e.g., Etheridge, 1983), the storage
for serpentine in the forearc mantle (Hyndman and Peacock, 2003),
the dewatering through the accretionary prism that depends on
its shape (Bekins and Dreiss, 1992), and the amount of high-pressure
hydrous minerals such as phase A in the subducting slab (e.g.,
Komabayashi et al., 2004).
Because of these factors that control the hydration/dehydration reactions of the slab and the escape of the fluids to the surroundings all
contribute to the magnitude of λ, it is not surprising that λ and thus
Δσ are not simply related to the plate tectonic parameters, such as the
convergence rate or age of the plate. Consequently, the seismic coupling
might not be simply related to the plate tectonic parameters.
If the average value of λ over the megathrust (Seno, 2009; Wang and
He, 1999) were available, it would provide a useful tool to differentiate
zones of future Mw ≥ 9 earthquakes from those where only Mw b 9
earthquakes are anticipated, when estimation of Δσ is not available as
in Ryukyu, Bonin and Tonga (Fig. 2). In such subduction zone segments,
quantification of the seismic coupling is usually insufficient (e.g., Ando
et al., 2009).
6. Conclusions
In this study, I propose a hypothesis to differentiate zones that possibly produce Mw ≥ 9 earthquakes from those that do not. I calculate Δσ
for Mw ≥ 7 earthquakes, by compiling the well-constrained seismic slip
distribution studies over worldwide subduction zones. Earthquakes are
grouped into class 1: Mw ≥ 9 earthquakes, class 2: Mw b 9 earthquakes
in a subduction zone segment in which at least one Mw ≥ 9 earthquake
occurred, and class 3: earthquakes in a subduction zone segment in
which no Mw ≥ 9 earthquake has occurred. The Δσ values of the
class 1, 2, and 3 events are 4.6, 3.4 and 1.6 MPa, respectively. The Δσ
values of earthquakes in each subduction zone are consistent with the
above averages; the Δσ values of the class 2 zones are greater than
3.2 MPa, and those of the class 3 zones are less than 1.7 MPa. Kuril–Hokkaido, which is a class 3 zone, is an exception, whose Δσ is 3.2 MPa, significantly greater than the Δσ values of other class 3 zones. The
pervasive Holocene tsunami deposits in SE Hokkaido suggest that this
segment may belong to class 2. Based on these observations, I propose
a hypothesis that if Δσ is greater than 3 MPa in a subduction zone
segment, this segment possibly produces Mw ≥ 9 earthquakes, and if
Δσ is less than 2 MPa, it would not.
I examine the scaling relations of the fault parameters, D, W, L, and
Mo, obtained in this study. Assuming that the shear stress is acting
only over the asperities, I obtain Δσ = ΔσASA/S and Δσ3S ∝ D2, the
latter of which is the scaling relation between D and S, when Δσ is
not constant. Based on this relation, I obtain the relation LogL =
0.5Log(Mo/Δσ1.5) + 1.85, and apply this to estimate the magnitude
for an earthquake that might rupture the entire Nankai–Suruga Trough,
and obtain Mw = 8.6–8.4 using Δσ = 1.0–0.7 MPa.
Provided that ΔσA does not vary much over the depths of subduction
zone earthquakes, Δσ represents SA/S, a fraction of the asperity areas in
the total fault area. This implies that the class 3 events have smaller asperities in a unit area of the plate interface and may fail to be triggered
to class 1 events by ruptures of smaller asperities. On the other hand,
ruptures of the class 2 events, which have the Δσ values similar to the
class 1 events, might be obstructed by barriers.
I show that 1 − λ ∝ Δσ, where λ is the pore fluid pressure ratio. This
relation holds roughly for six subduction zones for which Seno (2009)
obtained λ. This relation means that 1 − λ can be used as a proxy for
Δσ. The value of λ represents the hydrological condition at the thrust
governed by the fluid interaction between the fault gauge and the surrounding materials over a long geologic time. Because many factors related to the hydration/dehydration reactions of the slab and the escape
of the fluids to the surroundings contribute to the extent of λ, it is not
surprising that λ and thus Δσ are not simply related to the plate tectonic
parameters. If λ is estimated over the megathrust using the geophysical
methods, it could supplement estimation of Δσ for predicting the
zones of future occurrence of Mw ≥ 9 earthquakes.
Supplementary data to this article can be found online at http://dx.
doi.org/10.1016/j.tecto.2014.02.016.
Acknowledgments
I thank Seth Stein and the anonymous reviewer for their critical review of the manuscript, which helps much to improve it. I also thank
Tomowo Hirasara who taught me about the elliptic formula a long
time ago, without which I could not accomplish this work. Naoyuki
Kato and Hiroe Miyake's comments are helpful.
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