1972, Phys. Earth Planet. Interiors 6, 263—278. North-Holland Publishing Company, Amsterdam
MOMENTS, SOURCE DIMENSIONS AND STRESS DROPS
OF SHALLOW-FOCUS EARTHQUAKES IN THE TONIA-KERMADEC ARC*
PETER MOLNAR
Institute of Geophysics and Planetary Physics, University of California at San Diego, La Jolla, Calif 92073.
U.S.A.
and
MAX WYSS
La,nont—Doherty Geological Observatory of Columbia University, Palisades, N.Y. 10964, U.S.A.
Estimates of moments, source dimensions and stress drops were
made for 34 shallow-focus earthquakes in the Tonga arc between
1963 and 1969. Moments were determined from mantle wave
spectra and from long period P and S wave spectra. Source dimensions were estimated from the corner frequency of the P and
S wave spectra using BRUNE’s (1970) theory. From the moment
and the source dimension, the stress drop was approximately
determined. Source dimensions and the stress drops increase
with moment, and stress drops are typically between a few bars
and a few tens of bars for earthquakes with body wave magni-
1. Introduction
tudes of about 5~to 7. In addition, the data suggest regional
variations in stress drops. The earthquakes with relatively large
stress drops did not occur along the main underthrusting zone of
the arc that defines the boundary between the Pacific and Australian plates. These earthquakes, instead, reflect deformation
within one plate of lithosphere, and not movement along preexisting faults that separate two distinct plates. Thus, some of
these earthquakes may result from the creation of new faults.
High stresses might be expected for such earthquakes because
of the high strength of unfaulted rock.
shallow dipping fault planes (IsAcKs et at., 1969;
and MOLNAR, 1972). Abundant evidence from
seismicity (SYKES, 1966), from fault plane solutions
(ISAcK5 et at, 1969), and from relative velocities and
attenuation of seismic waves (e.g., BARAZANGI and
IsAcks, 1971; MITRONOVAS and ISACKS, 1971; OLIVER
and ISACKS, 1967) indicates that a slab of lithosphere
descends beneath the arc and is defined by the seismic
zone. SYKES (1966) noted, however, that at the northern
end of the arc, the belt of shallow activity bends westward. The fault plane solutions for shallow earthquakes
there are different from those along the main arc and
reflect primarily dip—slip motion on nearly vertical,
easterly striking faults with the south side moving
down (ISAcKS et a!., 1969; JOHNSON and MOLNAR,
1972). Therefore these events do not represent movement of one plate past another, but the formation of a
hinge fault within the Pacific plate. ISACKS et at. (1969)
predicted that higher stresses might exist in this region
because new faults are being created there. This movement would not always result from slip on preexisting
zones of weakness as in the case of earthquakes on
plate boundaries.
JOHNSON and MOLNAR (1972) determined fault plane
solutions for five other earthquakes that were different
JOHNSON
Using body and surface wave spectra, we estimated
moments, source dimensions, average dislocations and
stress drops of 34 shallow-focus earthquakes in the
Tonga—Kermadec arc (fig. I). The purpose was to
evaluate these parameters for earthquakes with magnitudes between about 5-i- and 7, to investigate the variation of these parameters for earthquakes in different
portions of the arc where tectonic conditions are different, and to compare shallow earthquakes occurring on
a lithospheric plate boundary with those occurring
within the plates themselves. The Tonga—Kermadec
arc is a particularly good region for such a study, because it is a very active region for shallow earthquakes
(SYKES, 1966), and because the tectonics of the region
are relatively simple and well understood (ISACKS et at.,
1969; JOHNSON and MOLNAR, 1972).
Most of the shallow focus seismicity is confined to
the region between the trench axis and the volcanoes,
This belt of activity defines the common boundary of
the Pacific and Australian plates. Focal mechanisms
of nearly all of these earthquakes indicate that the Pacific Ocean floor underthrusts the arc to the west along
*
Lamont—Doherty Contribution no. 1892.
263
264
P. MOLNAR AND M. WYSS
150’
the trench within the Pacific plate. They show that
175’A/
I
the principal compressive stress is approximately horizontal and perpendicular to the arc. All five of these
events also could have occurred in unfaulted rock and,
SAMOA IS
35
-
26
therefore, not on preexisting zones of weakness. Thus
15
23
the Tonga arc offers an excellent opportunity to compare earthquakes occurring on a plate boundary with
~I1
FIJL ISLANDS
-
•
A
e
B
others occurring within one plate, and therefore possibly not on pre-existing faults.
20’
2
28.
S E
ample,
TsuJIuRA
(1969),
(1970a)of
and
WYSSbody
and
BRUNE (1971)
showed
thatWvss
the spectra
seismic
o
waves vary notably as a function of depth and from
one region to another. This variation reflects differences
c22
•23
25’
~
34•
5.8••12
8 2—%• 35
Several recent studies indicate that one parameter,
such as magnitude is insufficient to describe and cornpare earthquakes except in a very general way. For ex-
in the physical parameters that describe the earthquake
source such as rupture area, average displacement,
stress drop, etc.
Several investigations indicate that at long periods
the body wave spectrum asymptotically approaches a
constant value, so that over a large portion the spec52
~
,~
31.
9
I
Fig. I. Map showing the earthquakes studied. The contour of
the Tonga—Kermadec Trench is at 6000 m. The symbols indicate
different types offault plane solution: A — shallow dipping underthrusts of the Pacific sea floor beneath the Tonga arc; B — the
hinge faulting of the Pacific plate; C — normal faulting beneath
the trench axis; D — thrust faulting east of the Tonga trench;
E — strike—slip faulting within the Australian plate (see ISACKS
et al., 1969; JOHNSON and MOLNAR, 1972). The stress drops in
bars are written next to the location of each event,
from either of the types mentioned above. One event,
west of the volcanoes showed strike—slip motion. This
event probably reflects internal deformation of the
overlying Australian plate. Solutions for two events
near the trench axis are normal faults with the principal
axis of extension nearly horizontal and perpendicular
to the trench. These events result from the bending
down of the Pacific plate as it underthrusts the arc and
indicate a stretching of the top of the bent plate
(STAUDER, 1968). Two unusual events occurred east of
trum is essentially flat (e.g., BERCKHEMER and JACOB,
1968; BRUNE and TUCKER, 1970). At high frequencies,
the spectrum appears to decrease at some power y of
frequency, usually y = 2. The corner frequency .f~is
defined as that frequency where the low frequency
trend of the spectrum, when extrapolated to higher frequency, intersects the trend of the high frequency portion, when extrapolated to lower frequency. Thus, at
least three independent parameters, the long period
level, f~and y, must be used to describe the spectrum
or time signal (see also HANKS and THATCHER, 1972).
Abundant evidence exists to show that events with
.
similar long period levels can have different corner
frequencies and vice versa.
Recent advances in seismic source theory show that
certain physical parameters (seismic moment and
source dimensions) describing an earthquake may be
determined from the spectra of seismic waves. AK1
(1966, 1967) demonstrated that the moment M0 is proportional to the flat, long period portion of spectrum,
and that
M
A~
—
°
—
I)
~
where jt is the shear modulus, A is the fault area, and ~i
is the average displacement on the fault. Numerous
SHALLOW-FOCUS EARTHQUAKES IN
TABLE
THE
265
TONGA—KERMADEC ARC
I
Pertinent data for the earthquakes studied. Data for events with Q of quality A were considered more reliable than those with quality
B; either more data were used or the individual spectra agreed better with one another for those with quality A than with B
DATE
LAT
Os
LONG
DEPTH
°w
MO M E N T (x 1025 dyne-cm)
SURFACE
26 Mar 63
31 Mar 63
(05)
31 Mar 63
(19)
20 May t~
30 Jul 63
8 Oct 63
31 Oct 63
18 Dec 63
29.7
29.9
177.8
177.7
20
48
1’40. (1)
7.5(2)
30.2
177.6
40
1.5(2)
30.8
29.8
15.0
21.8
24.7
178.2
177.2
173.2
174.8
176.7
39
50
25
32
44
P
93.
ii.
S
AV,
(5)
(5)
106.
9.6
3.5(4)
14.5(3)
20.2(8)
8.0(2)
7.7(7)
2.0(2)
4.7(3)
3.9(3)
4.6(4)
370. (1) 180. (2)
f
0(11xlO)
P
S
r (km)
P
S
A
A’!.
~2
U
Ac Quality
cm
bars
31.
10.
14,3
A
A
0.77(5)
1.4(5)
38.9
21.4
38.9
21.4
4750.
1370.
2.5
2.0(4)
15.0
15.0
706.
5.1
3.2
B
18.
7.8
1.9(2)
2.9
1.8(2)
3.7
270.
1.6(6)
1.6(7)
2.0(3)
1.6(4)
0.8(1)
18.7
18.7
1.1(2) 15.0
1.2(2) 18.7
37.4
18.7
18.7
15.3
18.0
37.4
1100.
1100
735.
1020.
4390.
23.
10.
5,L+
5.2
89.
12.
5.2
3.5
2.8
23.
A
A
B
A
B
11.
A
15.8
16.6
9 Jul 64
23.3
175.6
48
7.2(3)
5.7(3)
7.4(4)
6.9
1.9(3)
1.3(4) 15.7
13.4
14.2
6314.
15.
22 Mar 65
15.1
173.4
0
6.4(2)
9.8(5)
6.8(7)
7.6
1.8(5)
1.2(7) 16.6
14.5
15.’4
7145.
15.
11 Jul 66
20 Aug 66
19.3
23.6
173.4
176.0
51
‘40
1.0(1)
2.1(2)
1.4(3)
1.8(3)
0.8(2)
2.1(3)
1.1
2.0
2.0(3)
2.0(3)
1.3(2) 15.0
1.5(2) 15.0
13.4
11.6
14.4
13.6
651.
581.
1
17
5
14
22
12
27
29
Jan
Feb
Jun
Jun
Jul
Nov
Dec
Dec
67
67
67
67
67
67
67
67
15.2
23.7
21.3
15.2
33.5
17.2
22.3
22.8
173.8
175.2
174.5
173.6
179.0
172.0
174.8
175.3
0
19
33
11
39
3’s
33
3D
12. (2)
7.5(5)
2.7(4)
2.6(5)
3.0(1)
4.6(3)
7.9(3)
0.8(3)
‘4.8(5)
7.3(5)
1.6(2)
2.0(3)
1.9(6)
3.9(5)
8.6(5)
1.1(6)
2.4(4)
2.4(2)
1.7(4)
2.6(5)
2.6(5)
1.7(2)
2.3(3)
1.7(5)
1.7(5)
1.4(10)
1.7(6)
2.0(4) 11.5
1.7(2) 11.5
1.1(4) 17.6
13.0
1.2(2) 17.6
1.3(4) 17.6
1.0(5) 21.14
1.5(1) 17.6
8.7
10.2
15.8
1.2(1)
4.7(4)
14.1(4)
1.2(1)
6.2
6.8
2.3
2.4
2.0
4.14
7.1
1.0
10.3
11.1
17.0
13.0
16.7
15.2
20.1
16.7
19
20
25
26
15
20
30
25
29
26
11
29
30
29
Mar
Apr
Apr
Apr
May
May
May
Jul
Jul
Sep
Jan
Jan
May
Jun
68
68
68
68
68
68
68
68
68
68
69
69
69
69
17.14
15.7
15.2
15.3
29.7
30.7
30.9
30.7
22.4
30.5
28.4
17.2
32.2
30.5
172.8
172.6
173.1
173.1
179.0
178.3
177.6
178.3
174.9
178.1
176.8
171.5
178.1
178.2
33
30
33
33
33
45
41
60
33
33
68
33
34
43
1.5(4)
4.2(5)
1.5(1)
3.2(1)
18. (2)
27. (3)
4.8(4)
1.8(2)
1.7(3)
2.7(5) 4.8(3)
2.3(4) 1.7(2)
3.0(3) 1.6(1)
8.9(2) 23. (3)
40.(l4) 39. (3)
4.4(5) 7.6(2)
98.(1O)
2.6(5) 1.7(3)
53(10) 93. (3)
2.1(5)
2.4(3) 1.9(2)
2.2(5) 2.3(1)
2.1(4) 3.0(2)
1.6
3.9
1.9
2.8
18.
36.
5.0
98.
3.1
63.
2.0
1.3
2.1
1.5
2.1(2)
2.0(5)
1.7(4)
3.2(3)
2.1(2)
1.0(13)
1.5(6)
0.8(9)
1.9(5)
1.0(9)
1.6(5)
4.0(3)
2.1(4)
2.8(5)
1.2(3)
1.5(3)
2.0(2)
2.1(1)
2.1(2)
0.9(3)
1.2(2)
14.5
11.6
8.8
8.3
8.3
19.4
14.5
l’4J+
13.7
14.7
9.2
11.3
27.9
18.5
37.4
15.2
28.4
18.7
7.7
14.3
10.4
4.3(3)
65. (2)
1.8(1)
1.7(5)
1.6(1)
1.1(7)
studies show that M0 may be reliably determined from
spectra of surface waves (e.g., AK!, 1966; T5AI and
AKI, 1969) and of body waves (e.g., HANKS and WYSS,
1972, W~ssand HANKS, 1972a, b).
BRUNE (1970) proposed that, for S waves, the source
dimensions are proportional to the corner frequency
and in the case of complete stress drop that y = 2. He
considered a circular fault and obtained for the radius
r of the fault
114.5
23.8
7.9
14.5
8.7
9.1
A
2.5
5.0
1.6
3.5
B
A
333.
386.
908.
531.
887.
726.
1270.
887.
27.
25.
3.6
6.6
3.3
8.6
7.9
1.6
25.
22.
2.0
‘4.9
1.9
5.5
3.8
1.0
A
A
B
B
A
A
A
A
651.
589.
678.
266.
1400.
2440.
1070.
‘+390.
726.
2530.
1100.
186.
642.
3413.
3.5
9.5
14.0
15.
64.
21.
6.7
32.
6.1
35.
2.5
14.
4.6
63
2.3
6.6
2.6
15.
52.
7.3
3.5
8.2
3.9
12.
1.3
17.
3.1
58
B
A
B
B
B
A
A
A
B
A
B
B
B
A
as many of those considered here, show that Brune’s
theory extended to include P waves is reliable (HANKS
and Wyss, 1972; W~ssand HANKS, 1972a, b).
2. Analysis
For all events considered, the radiation pattern was
known from published fault plane solutions (ISACKs
et a!., 1969; JOHNSON and MOLNAR, 1972).
We analyzed both body and surface waves using data
(2)
from the
World-Wide
Standardized
Seismograph
Network
(WWSSN).
Recordings
with clear
Love or Rayleigh waves were digitized. From the spectral density at
where /3 is the shear wave velocity andf0 is the corner
frequency. For P waves, W~sset at. (1971) replaced /3
with the P wave velocity. Experimental data for four
earthquakes, with approximately the same magnitude
100 or 150 s, the fault plane solution and the depth of
focus we determined moments (table 1) using the tables
given by BEN-MENAHEM et at. (1970).
Our analysis of body waves was essentially the same
‘.
~,
=
22 3/3_~
itfo
14.2
15.0
17.6
9.5
14.2
28.9
19.9
37.4
1.2(3) 15.7
0.73(3)29.9
18.7
2.2(2)
7.5
1.2(1) 114.2
2.0(1) 10.7
14.5
13.14
17.4
11.6
7.9
266
P. MOLNAR AND M. WYSS
procedure as that of HANKS and WY55 (1972) and WYSS
and HANKS (1972a, b). For each earthquake, we digitized P and S waves recorded by both short period and
long period instruments from the WWSSN. By using
both short period and long period seismograms we increase the band width or the spectrum, and thereby
gain an advantage over studies conducted in the time
domain from either short or long periods alone. The
is the value of the flat portion of the displacement spectral density at low frequency, corrected for attenuation
and divided by 2.5 to correct the crustal transfer function. Moments were estimated for each signal. Because
only the low frequency portion of the spectrum is used
in this calculation, the correction for attenuation did
not significantly alter the estimates of the moment. For
small values of R04 (< 0.1), small uncertainties in the
digitized signals were interpolated to give uniform
sample spacing. These signals were plotted and cornpared with the original records to insure that no errors
in digitization were made. Although exclusion of pP
and sP was impossible, window lengths were chosen to
avoid contamination by prominent later phases. Using
the fast Fourier transform algorithm, the spectrum was
determined. A correction was then made for the effect
of the instrument response to obtain the spectrum of
the ground motion at the recording site.
Two spectra were then plotted on a log—log scale:
me is the spectrum of the ground motion and the
fault plane solution (a few degrees) lead to uncertainties in R04, and therefore in M0, as large as an order of
magnitude. Therefore, in such cases we did not use
these values of the moment.
An average moment from P waves and from S waves
was determined for each event by weighting each estimate equally (table 1). The average moment for each
other is the spectrum corrected for the effect of attenua-
a horizontal straight line was drawn by eye through the
tion from source to receiver using the average Q values
of JULIAN and ANDERSON (1968) and assuming that Q
is independent of frequency. These spectra comprise
the basic data used.
No attempt was made to remove the effect of the individual crustal transfer function beneath each recording site. BERCKHEMER and JACOB (1968), however,
showed a few examples indicating that its effect is not
large. For all earthquakes, the records of several stations were used. In this way, irregularities in the mdividual spectra caused by the local crustal structure
were
reduced.
In thebycalculations
of for
moments,
the
spectrum
was divided
2.5 to account
the average
effect of the crustal transfer function (BEN-MENAHEM
et a!., 1965).
The determination of the moment from body waves
follows from
long period portion of the spectrum. Another straight
line was fit by eye through the high frequency portion
of the spectrum, and the intersection of these lines gives
the corner frequency. No attempt was made to constrain
the fall-off at high frequencies, y, to be equal to two.
Corner frequencies for P and S were averaged independently, and using eq. (2), radii of equivalent circular faults, r, were determined from each average
corner frequency. An average radius was then obtained
by weighting those derived from P and from S proportionally to the number of2, signals
usedthefor
each. The
and using
moment
and
area
was
assumed
to
be
itr
eq. (1) the average displacement was estimated. Manipulation of equations derived by KEILIS-BOROK (1959)
shows that the stress drop can be estimated from the
moment and the radius,
=
3RQ(0)
4itpvR
(3)
04
(e.g., KEILiS-B0R0K, 1960), where p and v are the density and velocity (~
or /3) of the material at the source.
event was calculated by weighting each estimate from
body waves equally but by weighting estimates from
surface waves twice as much, because these were considered to be more reliable.
Todeterminethecornerfrequencyfromthespectrum,
7
Aa=-
3
1--~M0/r
.
(4)
These results are given in table I.
Since there is a certain amount of subjectivity involved in determining corner frequencies and moments, it
is necessary to show the spectra used. Showing each
R
04 is the radiation pattern obtained from equations in
et a!. (1965) and from the fault plane
solution. R is the correction for geometrical spreading
and is taken from JULIAN and ANDERSON (1968). 2(0)
BEN-MENAHEM
spectrum individually, however, would require too
much space. Instead, in the Appendix we present average spectra (average of all available stations) for P and
for S from each event studied. Brune’s theory con-
SHALLOW-FOCUS EARTHQUAKES IN THE TONGA—KERMADEC ARC
I
I
•
I
I
IILL~
I
I
I
11111
I
I
I
PWAVES
I
267
-
A S WAVES
1027
•
A
-
I
uJ
>
60
•
>-
60
A
C
I—
26
.
LU
A
C
.
.
.
A
A
A
1025
,
iii
io~
Fig. 2.
A
I
;~26
I
I
iiiil
I
1027
MOMENT (SURFACE WAVES)
Moments determined from body waves as a function of the moment determined from surface waves. The straight line
represents the idealized case of equality of these determinations.
siders a root mean square average spectrum that is
radiated by the source. To calculate such an average
spectrum it is necessary to correct each observed spectrum for the effects of radiation pattern and geometrical spreading. This is accomplished by multiplying each
spectrum by the weighting factor M0/Q(0) appropriate
for each. The spectra Q (j) in the Appendix are
/
N
I
=
~
M’
(~Q
~—
j
\2]+
( ) 1(J))
(5)
where N is the number of stations used. These spectra
are thus given in units of dyne cm, and the values of
~(f) at low frequencies determine the average moment.
Thus the reader may estimate the moment as well as
the corner frequency directly from the figures. Our
choices are shown by the dot in each plot.
It is necessary to discuss the uncertainties of the calculations as they are not small. We think that the moment can be reliably determined within a factor of
about two both from body and from surface waves,
The individual moments determined from each wave
type usually agree within a factor of two and the aver-
age values for each wave type are generally within a
factor of two of each other. Fig. 2 shows the moments
determined from P and S waves as a function of the
moment determined from surface waves. The agreement between them is good. Moreover, previous studies
(e.g., HANKS and W~ss,1972; Wvss and HANKS, l972a,
b) indicate agreement within a factor of two between
the moments determined from seismic spectra and
those
from
We determined
assumed that
all field
of thedata.
hypocenters were in the
mantle, rather than in the crust. If this is incorrect, because of the smaller ~t in the crust, the moments should
be reduced by about a factor of 2. The stress drops,
however, are essentially unchanged, because the smaller
velocity in the crust reduces the values of radii (eq. 2).
Uncertainties of f~and y are more difficult to estimate. We assumed that Q is independent of frequency
and used the average Q of JULIAN and ANDERSON
(1968), which is based on surface wave data with longer
periods than generally considered here. The assumption
that Q is independent of frequency has been questioned
(JACKSON and ANDERSON, 1970) and some evidence sug-
268
P. MOLNAR AND M. WYSS
gests that Q increases with frequency at least over the
bands studies (ARCHAMBEAUet al., 1969; FEDOTOV and
BOLDYREV, 1969; SOLOMON, 1971). Thus the correction
for attenuation
is probably a maximum estimate. The
value of y depends strongly on the attenuation
correction as may be seen by comparing Q corrected and uncorrected spectra in the Appendix. The uncertainty
in
the Q model makes it impossible to estimate y accurately, however. In the spectra corrected for attenuation,
y is usually approximately
two, and therefore these
spectra are used in the calculation
of source parameters. It is important
to note, however, that in most
cases the corner frequencies determined
from the Q
corrected and uncorrected
spectra agreed with one
another within about 20”/,. Thus, the attenuation
correction does not, in general, affect the determination
of
the corner frequency.
We think that the corner frequency for an individual
spectrum can be estimated within a factor of about ,/2.
Usually, corner frequencies from different signals for a
single earthquake are within this factor of one another.
Because recordings at teleseismic distances sample only
a small portion of the focal sphere, it is not possible to
average the spectra over the entire radiation pattern
as specified in Brune’s theory. Rupture propagation
can focus energy in different directions. On the other
hand, estimates of rupture velocities and analysis of
spectra suggest that the effect on the corner frequency
of focusing due to rupture propagation
is not very important for P waves, and probably is not usually important for S waves (BERCKHEMERand JACOB, 1968 ;
FUKAO, 1970; MIKUMO, 1969, 1971a, b). More importantly, HANKS and WYSS (1972) and WYSS and HANKS
(1972a, b) found agreement within a factor of ,/2 between the fault dimension determined from corner frequencies and from field observations for four well-studied events. These results offer strong support for the
application
of BRUNE’S (1970) theory to events recorded at teleseismic distances.
The stress drop depends upon the geometry of the
fault plane, so that sources with the same moment and
the same fault area can have different stress drops (e.g.,
CHINNERY, 1969a). The assumption that the fault plane
is essentially circular would of course be invalid for very
large earthquakes
or for moderate size earthquakes on
long thin faults such as the San Andreas fault. In an
island arc, however, where the fault zone between two
plates is more than 100 km wide, it is reasonable
to
assume that rupture areas are essentially circular. In
fact, SYKES (1971) showed that in the Aleutians, earthquakes with magnitudes less than 7+ usually had rupture zones that were approximately
square (or circular)
and not rectangular
as for much larger events. The
assumption
of a circular fault is probably
reasonable.
Hence with an uncertainty
of a factor of two in the
moment and of ,/2 in the corner frequency, from eq.
(4) the products of these uncertainties
implies that the
maximum error in the stress drop should be a factor of
about six. In one sense, this uncertainty
is conservative; T. C. Hanks (personal communication)
pointed
out that for a given spectrum the determinations
of
corner frequency and moment are not independent.
An
underestimate
in the moment by a factor of 2 leads to
an overestimate in the corner frequency by a factor of
42. If the only uncertainties
in the analysis were in
determining
the corner frequency and the flat portion
of a reliably measured average spectrum radiated from
the source, the uncertainty
in the stress drop would be
only 42. A more realistic estimate of this uncertainty
is probably a factor of 3 for the better determination
of stress drop and larger for the poorer determination.
3. Results
Table 1 lists the events studied and the estimated
moments, corner frequencies, radii of equivalent sources, average displacements
and stress drops. In fig. 3,
radii, average displacements,
and stress drops are plotted as a function of moment. Different symbols are
used for earthquakes
with different types of faulting,
following the pattern in fig. 1. Let us first consider the
events that occur on the plate boundary,
those indicated by solid symbols.
The radii (or source dimensions)
increase as the
moment increases, a phenomenon
observed in all previous studies (e.g. TOCHER, 1958; LIEBERMANN and
POMEROY, 1970). Moreover, these values of radii and
therefore of fault areas agree well with other studies of
fault dimension of earthquakes
with the same magnitude or moment (e.g., CHINNERY, 1969b). Similarly the
average displacement increases with moment, as noted
by CHINNERY (1969b), KING and KNOPOFF (1968) and
others, and the values of displacement
also agree well
with those tabulated by the authors for events of about
269
SHALLOW-FOCUS EARTHQUAkES IN THE TONGA—KERMADEC ARC
moment, however, do not describe well the trends in
100
•
~l0
fig. 3. Although this difference could reflect errors in
either the analysis or the theory, for earthquakes in the
magnitude range, 5~-to 7, all studies show a considerable variation in fault dimensions (e.g., LIEBERMANN
and POMEROY, 1970). In fact, the relationships of fault
dimension and moment are very different for events
less than about 5+ compared with events larger than
about 7. Moreover, the data used in these earlier studies is primarily from long thin rectangular faults for
•
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100
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•
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25dyne-coU 00
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MOMENT lxlO
Fig. 3. Radii ofequivalent sources (top), average displacements
(middle) and stress drops (bottom) as a function of moment. The
symbols are the same as in fig. 1. The dotted line is from WYSS
(l970b).
the same size. These results give us confidence that
such parameters were reliably determined,
CHINNERY (1 969b) gave empirical relationships between magnitudes and moment, fault area, average displacement, etc. These expressions when combined to
give fault areas versus moment or displacement versus
which
theofrelationship
source
a
function
magnitude between
is different
fromparameters
that for a as
circular fault. Hence, we do not consider the different
trends between the data in fig. 3 and those given by
CHINNERY (1969b) to be significant.
The stress drops of earthquakes on the plate boundary also increase with magnitude, a result suggested
previously by BERCKHEMER and JACOB (1968); KING
and KNOPOFF (1968) and WYSS (l970b). Shown in
fig. 3 is the line given by WYSS (1970b) that approximately relates stress drop to moment. This line is parallel
to the trend of the data but gives stress drops about a
factor of three higher than the observed values. This
difference may indicate a systematic difference in the
stress drops in the Tonga—Kermadec arc from those
considered by Wyss. More likely, however, the difference is due partly to the different methods of calculation of stress drop and to the different fault geometries
of the events considered here from the primarily long
thin rectangular slip faults considered by Wyss. Because of the uncertainty in the stress drops, the deviations from his data are not considered significant.
AK! (1972) has challenged the idea that stress drop
increases with magnitude, and if all events in fig. 3 are
considered, the data do not clearly indicate such an increase.
Onedrop
possible
these data
the
stress
doesinterpretation
increase withofmoment,
butis that
also
has strong regional variations. W~ssand BRUNE (1971)
and THATCHER (1972) interpreted spectral differences
.
.
.
as evidence for possible regional variations in apparent
stresses or stress drops in western North America.
When earthquakes only on the San Andreas fault are
considered, however, the stress drops appear to increase with moment (KING and KNOPOFF, 1968; WYSS,
l970b). Thus it is important to compare earthquakes
from the same region and in such cases the stress drop
does appear to increase with the moment.
270
P. MOLNAR AND M. WYSS
The five events with the smallest source dimension
but not smallest moments occurred not on the plate
boundary, where a well developed zone of weakness
exists, but within one of the plates. Two of these events
occurred at the north end of the arc where plate tec-
tonics requires that a new fault is forming (ISACKS et at.,
1969). The other three events appear to result from a
stress within one of the plates (JOHNSON and MOLNAR,
1972) and could also reflect the creation of new faults,
The stresses required to break unfaulted rock are generally much larger than those necessary only to overcome
friction on a preexisting zone of weakness. Thus, if
large stress drops imply high strength, the large stress
sphere is important because of the greater potential
destruction of high stress drop earthquakes. In fact,
from the much larger area over which the 1886 Charleston earthquake was felt compared with the 1906
San Francisco earthquake, REID (1910) suggested that
higher stresses existed in the eastern United States than
in the west. Much of the damage caused by earthquakes
is from shaking with frequencies near about 1 Hz. A
large event rich in the high frequency portion of the
spectrum (a high stress drop earthquake) will therefore
be more destructive than a low stress drop event. More-
ing. Although the uncertainties in the stress drops are
nearly as large as the differences in stress drops between these two groups, we conclude from the consistent pattern that these differences are real.
Not all of the events occurring within plates (partially open symbols in fig. 3) had large stress drops.
This is also reasonable, however, as many of these
events probably did occur on preexisting faults, For
instance, surely some of the events showing hinge faulting result from the movement of one piece of the Pacific plate past another without actually extending the
fault zone.
over, because attenuation appears to be lower within
the plates than at their margins (e.g., MOLNAR and
OLIVER, 1969), the higher frequencies will persist for
longer periods of time in the more stable regions. Thus,
although infrequent, earthquakes within plates of
lithosphere should not be ignored in a consideration of
earthquake risk.
Also, because the principal seismic criterion for
identifying underground nuclear explosions is by a
comparison of the long period and short period portions of the spectrum, high stress drop earthquakes are
the most likely events to be misidentified. It is important to recognize that such events may often occur
within plates of lithosphere and are perhaps less likely
along well defined plate boundaries.
4. Discussion
Acknowledgments
Moments, source dimensions, average dislocations
and stress drops were determined from the analysis of
body and surface wave spectra for 34 shallow earthquakes in the Tonga—Kermadec arc with magnitudes
between about 5+ and 7. Stress drops range from a few
bars to a few tens of bars, and for events along the
boundary between the Pacific and Australian plates
they increase with increasing moment. Also, as found
in western North America (THATCHER, 1972; W~ssand
BRUNE, 1971), a significant regional variation in stress
drop is indicated. Some events that occur within either
the Pacific or Australian plates have higher stress drops
than events with the same moment that occur on the
boundary of these two plates. The events within the
plates may result partly from the creation of new faults
and thus require higher stresses than other events that
result simply from movement on preexisting zones of
weaknesses.
That high stresses may exist within plates of litho-
We thank K. H. Jacob for useful discussions, and he,
T. Mikumo and P. Richards offered many valuable
suggestions for improvement of the manuscript.
This research was supported primarily by the National Science Foundation Grant GA-22709 at Lamont—Doherty Geological Observatory. P. Molnar’s
support at the Institute of Geophysics and Planetary
Physics, University of California, San Diego, was from
NSF-GA-l9355.
drops associated with these five events are not surpris-
References
AKi, K. (1966), Bull. Earthquake Res. Inst. Tokyo Univ. 44,
73—88.
AKI, K. (1967), J. Geophys. Res. 72, 1217—1231.
AK!, K. (1972), Tectonophys., to appear.
ARCHAMBEAU, C. B., E. A. FLINN and D. G. LAMBERT
(1969),
J. Geophys. Res. 84, 5825—5866.
BARAZANG!, M.
BEN-MENAHEM,
and B. ISACKS (1971),J. Geophys. Res.76, 8493—
A., S. W. SMITH AND T. -L. TENG (1965), Bull.
Seismol. Soc. Am. 55, 203—235.
SHALLOW-FOCUS EARTHQUAKES IN THE TONGA—KERMADEC ARC
A., M. ROSENMAN and D. G. HANKRIDER (1970),
Bull. Seismol. Soc. Am. 60, 1337—1388.
BERCKHEMER, H. and K. F-I. JACOB (1968), Investigation of the
BEN-MENAHEM,
dynamical process in earthquake foci by analyzing the pulse
shape of body waves, Final Sci. Rept. Contr. AF61 (052)—80I,
Inst. Meterol. Geophys., Univ. Frankfurt, 85 pp.
N. (1970), J. Geophys. Res. 75, 4997—5009.
BRUNE, J. N. and B. TUCKER (1970). In: Abstr. Progr. Ann.
Meeting Geol. Soc. Am., 506.
CHINNERY, M. A. (1969a), PubI. Dominion Obs. Ottawa 37
(7), 211—223.
BRUNE, 3.
CHINNERY, M. A. (1969b), Bull. Seismol. Soc. Am. 59, 1969—1 982.
FEDOTOV, S. A. and S. A. BOLDYREV (1969), Izv. Akad. Nauk
SSSR Ser. Fiz. Zemli (9), 17—34.
FUKAO, Y. (1970),
Bull.
Earthquake
Res. Inst. Tokyo Univ. 48,
707—727.
C. and W. THATCHER (1972), J. Geophys. Res. 77,
4393—4405.
HANKS, T. C. and M. Wyss, (1972) Bull. Seismol. Soc. Am., 62,
561—590.
IsAcKs, B., L. R. SYKES and J. OLIVER (1969), Bull. Geol. Soc.
HANKS, T.
Am. 80, 1143—1470.
JACKSON, D. D. and D. L. ANDERSON
(1970),
Rev. Geophys.
8,
1—64.
T. and P. MOLNAR (1972), J. Geophys. Res., 77,
5000—5032.
JuLIAN, R. and D. L. ANDERSON, (1968). Bull. Seismol. Soc. Am.
58, 339—368.
KEILI5-B0R0K, V. 1. (1957), Tr. Geofiz. Inst. Akad. Nauk SSSR
Sb. Statei 40, 201 (English Transl.: Soviet Res. Geophys. 4
(Am. Geophys. Union, 1960)).
KEILIS-BOROK, V. 1. (1959), Ann. Geofis. 12, 205—214.
KING, C. and L. KNOPOFF (1968), Bull Seismol. Soc. Am. 58,
JOHNSON,
249—258.
271
(1970), Bull. Seismol.
Soc. Am. 60, 879-890.
MIKUMO, T. (1969), J. Phys. Earth 17, 169—1 72.
MIKUMO, T. (1971a), J. Phys. Earth 19, 1—19.
MIKUMO, T. (1971b), Phys. Earth Planet. Interiors 6, 293—299
(this issue).
MITRONOVAS, W. and B. ISACKS (1971), J. Geophys. Res. 76,
7154—7180.
MOLNAR, P. and J. OLIVER (1969), J. Geophys. Res. 74, 2649—
2683.
OLIVER, J. and B. IsAcKs (1967), J. Geophys. Res. 72,4259—4276.
REID, H. F. (1910), The mechanics of the earthquake. In: The
California Earthquake of April 18, 1906, Rept. State Earthquake Invest. Comm. 2 (Carnegie Inst. Wash.) 192.
LIEBERMANN, R. C. and P. W. POMEROY
SOLOMON, S. (1971), Ph. D. Thesis, Massachusetts Inst. Technol.
STAUDER, W. M. (1968), J. Geophys. Res. 73, 3847—3858.
SYKES, L. R. (1966), J. Geophys. Res. 71,
SYKES, L. R. (1971), J. Geophys. Res. 76,
2981—3006.
8021—8041.
THATCHER, W. (1972). J. Geophys. Res. 77, 1549—1565.
TOCHER, D. (1958), Bull. Seismol. Soc. Am. 48, 147—153.
TSAI, Y. and K. AK! (1969), Bull. Seismol. Soc. Am. 59, 275—
288.
TSUJIURA, M. (1969), Bull. Earthquake Res. Inst. Tokyo Univ.
47, 613—633.
WySs, M. (1970a), J. Geophys. Res. 75, 1529—1544.
Wyss, M. (l970b), P~.D. THESIS, California Inst. Technol., 239.
WYSS, M. and J. N. BRUNE (1971), Bull. Seismol. Soc. Am. 61,
1153—1168.
Wyss, M. and T. C. HANKS (1972a), Geol. Survey Profess. Papers,
to appear.
Wyss, M. and T. C. HANKS (1972b), Bull. Seismol. Soc. Am.,
62, 591—602.
WYSS, M., T. C. HANKS and R. C. LIEBERMANN (1971), J.
Geophys. Res. 76, 2716—2729.
Continued on next page
272
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