Geophys. J . R.ustr. SOC.(1975) 41, 37-49.
Long-Term Behaviour of an Aftershock Sequence:
The Inangahua, New Zealand, Earthquake of 1968
Russell Robinson*, Walter J. Arabaszt and F. F. Evison
(Received 1974 December 3)$
Summary
The behaviour of the aftershock sequence of the Inangahua, New
Zealand, earthquake (magnitude 7. I, depth 12 km), as determined 3.6
years after the main event, can be compared with the behaviour during
the first 40 days of the sequence. The b-value (slope of the magnitudefrequency relationship) remains near unity and the rate of aftershock
occurrence is consistent with a decay proportional to (time)-''O5.
Epicentres of the late aftershocks (magnitudes less than 3.3) occupy
roughly the same area as the epicentres of the early aftershocks (magnitudes greater than 3 S), an elliptical region elongated along the trace of
the Glasgow Fault. No fault plane is defined by the hypocentres of the late
aftershocks. There has been a radical change in mechanisms from the
thrusting of the early aftershocks (and main event) to the normal faulting
mechanism of the late aftershocks. This change can be interpreted as
due to outflow of pore fluids if the dilatancy hypothesis of earthquake
occurrence is correct.
Introduction
The aftershock sequences of many large and moderate magnitude earthquakes
have been studied in detail and their main properties are now well known. Even for
large events, however, the magnitude range of the aftershocks soon falls to such low
levels that only seismographs designed to detect micro-earthquakes and situated
very close to the aftershock zone can supply information on the long-term behaviour
of the sequence. In view of recent proposals linking the earthquake phenomenon
with migrations of pore fluids, on a time scale which for large earthquakes runs into
many years, it is especially desirable to obtain a knowledge of such long-term
behaviour of aftershock sequences.
The essential features common to aftershock sequences as determined over periods
of up to a year are (e.g. Utsu 1961; Page 1968; Ranelli 1969): (1) the frequency of
occurrence at time t after the main shock is given by n(t) = At-p, where A is a constant
related to the main shock magnitude and p is near unity; (2) the frequency of
occurrence as a function of magnitude is given by n(M)where log n ( M ) = a + bM and
a and b are constants; (3) the aftershocks are closely related spatially to the region of
faulting in the main shock but the hypocentres may or may not delineate planar
fault surfaces; (4)most aftershocks display a focal mechanism similar to that of the
* Present address: Geophysics Division, Department Scientific & Industrial Research, Wellington.
t Present address: Department of Geological and Geophysical Sciences, University of Utah,
Salt Lake City, Utah 84112.
1: Received in original form 1974 August 16.
37
38
R. Robinson, W. J. Arabasz and F. F. Evison
FIG.1. Index map showing the northern part of the South Island of New Zealand.
The large dot indicates the epicentre of the Inangahua earthquake. Major faults
are also shown.
main shock although there may be considerable complexity in detail: ( 5 ) aftershock
sequences occur only at shallow depths (less than about 20 km deep).
The continuing aftershock activity of the 1968 May 23 Inangahua earthquake
(ML= 7.1) provided an opportunity to re-examine the characteristics of an aftershock sequence after a period of three and a half years. This is the only earthquake
of magnitude 7 or greater to have occurred in New Zealand since 1960, and the only
such event to have occurred on land for a much longer time. Adams, Eiby & Lowry
(1968) and Adams & Lowry (1971) have studied the main shock and the first 40
days of the aftershock sequence. The epicentre of the main shock was located at
41.77" S, 172.01" E, close to the surface trace of the Glasgow Fault (Fig. l), and a
focal depth of 12 km was assigned on the basis of identified crustal phases. Lensen
& Otway (1971) have analysed the surface deformation associated with the earthquake.
During the main event the movement was predominantly thrusting, with a small
left-lateral component, on the eastward-dipping Glasgow Fault.
The Glasgow Fault is one of several important breaks following the dominant
north-north-east trend of the region north-west of the Alpine-Wairau Fault (Fig. 1).
On the nearby White Creek Fault, a large thrust movement with a left-lateral component occurred during the 1929 Murchison earthquake (MLx 7 3/4). Lenson &
Suggate (1968) indicate that the Glasgow Fault has been active in post-glacial times
and that late Quaternary faulting has resulted in net uplift of the region. Seismically,
Long-term behaviour of an aftershock sequence
39
the Inangahua area lies at the southern end of the Main Seismic Region of New
Zealand (Eiby 1971), in which generally diffuse crustal activity occurs above a northwest-dipping Benioff zone of mantle earthquakes. The Benioff zone does not extend
quite as far south-westwards, however, as Inangahua.
Field procedure and data analysis
Four portable micro-earthquake recording systems (Sprengnether model MEQ600) were operated in networks involving a total of six sites in the Inangahua area
during the period January 4-19, 1972 (Fig. 4 and Table 1). Displacement magnifications ranged from 8 x lo5 to 3 x lo6 at the peak response frequency of 23 Hz. The
accuracy with which arrival times could be determined was f0.1 s for sharp arrivals
at a recording speed of 60 mm mn-l. Clocks were calibrated by recording radio time
signals at least once daily at each station.
For each earthquake the seismograms were read for the arrival times and
amplitudes of P and S phases, the event duration, the sense of first motion, and where
appropriate the nodal character. During periods of normal noise (< 0 - 5 mm),
it is believed that all earthquakes in which the trace amplitude (zero to peak) reached
2 mm could be recognized.
All sufficiently well-recorded events were located by an iterative procedure in
which the hypocentre is adjusted so that the travel-time residuals are minimized in a
least squares sense (Geiger 1912; Lee & Lahr 1971). Both P and 5‘ phases were used
when available, and depths could be restricted if desired. In all cases a second solution
was automatically computed using the ‘ three-parameter ’ method (James et al.
in which reliable S - P times are used to fix the origin time, leaving the three spatial
co-ordinates to be determined. The velocity model used was the standard New
Zealand crustal model (Hamilton 1966), consisting of two layers over a half space,
with layer thicknesses of 12 and 21 km, P velocities of 5.5, 6.5, and 8.1 km s-’, and
Poisson’s ratio 0.25. Successfully located events were assigned to one of four classes
Table 1
Station data
No.
1
2
3
4
5
6
Name
Lyell
RoughCreek
Fletcher Creek
Heaphy Mine
Mackley
Granity
Latitude
41°48.153’S
41 53.032
41 59.281
41 52.511
41 47.659
41 40.899
Longitude
172”05.713’E
171 57.100
171 49.597
171 50.884
171 54.356
171 55.170
Dates operating
1972 January 4-19
1972 January419
1972 January 4-15
1972 January 5-7
1972 January 8-19
1972 January 16-20
Table 2
Classes of location quality
Class
A
B
Number of phases
Standard error
60.T.
6 from 4 stations
GO.1 s
GO.1 s
6
G0.3
G0.3
a5
G0.3
G0.3
C
24
G0.4
G0.4
D
Standard Error = (C (travel-time residuals)2/
(number of phases - 1)*
60.T. = difference in origin time computed by the threeparameter and four-parameter methods (see
text).
40
R. Robinson, W. J. Arabasz and F. F. Evison
according to quality of solution, as shown in Table 2. Class C events were given
restricted depths, chosen by minimizing the standard error. Class D events were
excluded from the map and cross-sections of hypocentres.
Because of the small number of stations used in this study it is important to assess
the accuracy of the locations. Numerical tests involving the introduction of random
0 -1 s timing errors and plausible variations in the velocity model show that at best a
well-recorded event would be located with errors of 0.5 km in latitude and longitude,
1 a 0 km in depth, and 0.1 s in origin time. Location errors may sometimes approach
2 km in any direction because of reading errors or poor station geometry, but this is
unlikely for class A and B events. Substantial lateral changes in velocity possibly
could cause larger absolute errors in location but would not affect the relative location
accuracy to as great an extent. Very large errors (10 km) would be needed to substantially change the interpretation of the pattern of first motions.
The use of clear S arrivals is necessary for the determination of high quality
locations within a small network such as ours. The principal effect of an error in
picking S would be to alter the computed focal depth. Generally, however, errors
in reading the S phase became apparent during the location procedure, whether as
large standard errors, a large difference in the origin time as determined by the two
methods, or a zero depth with large computed error. Few class A or B events are
likely to be much in error from misreading the S phase.
Fhxptirde
FIG.2. Frequency-magnitude relation for micro-aftershocks at Inangahua 3.6
years after the main shock. Number of events is non-cumulative.
Table 3
Frequency-magnitude relation, Inangalma aftershock sequence
Time interval
(days)
Magnitude range
M O
2-5-40
1332-1347
3* 7 - 6 0
3.1-5.5
1.0-3.1
Number of events
b
248
449
173
0.93 +O. 13
1.06kO. 10
1* 01 k 0.15
41
Long-term behaviour of an aftershock sequence
The frequency-magnitude relation
The frequency of occurrence with respect to magnitude of the late aftershocks in
the Inangahua sequence, as expressed by the Gutenberg-Richter relationship
log n (M ) = a-bM,
(1)
has been determined from the events recorded at station 3, the most sensitive station.
Magnitudes were determined for all recorded events, many of which were not locatable,
from signal duration and S - P interval (Appendix). Here n(M)is the number of
events of magnitude between M and M + d M , and it is believed that all events have
been included for which M 2 1.0.
The data are plotted in Fig. 2; the summary in Table 3 allows comparison with the
results of Adams & Lowry (1971). The value of the constant b is determined in each
case by the maximum-likelihood method (e.g. Page 1968) and is the same as that
obtained by considering the relation between the cumulative number N ( M ) and M .
Numerically, the values in Table 3 are not significantly different. These results suggest
that at Inangahua the variation of frequency with magnitude remained nearly constant,
with b close to unity, from the beginning of the sequence until at least three-and-a-halfyears later, and over a magnitude range of five units.
Long-term decay of aftershock rate
In the expression for the frequency of occurrence of aftershocks with respect to
time,
n(t) = A t - p ,
(2)
Adams & Lowry (1971) found by a least-squares method applied to the first 40 days
of the Inangahua sequence, for all shocks of magnitude 3.8 and greater, that A = 30
and p = 1.05fO.06. We wish to examine the long-term decay of aftershock rate by
comparing their data with that obtained 3.6 years after the main shock.
I
I
I
I
I
I
Days offer main event
FIG. 3. Rate of aftershock activity (events/day) at Inangahua for events with
magnitude 2 1.0, us time (days) after the main shock. Circles represent data
from Adams & Lowry (1971); squares, data from earthquakes located by the
New Zealand Seismological Observatory for the latter part of 1968; the diamond,
the rate determined in this study.
42
R. Robinson, W. J. Arabasz and F. F. Evison
From a 12-day sample of activity at station 3, the rate of aftershock occurrence
during our study for events with magnitude >, 1 - 0 was 1 6 - 4 2 1 - 2events/day. In
order to compare this with the data of A d a m & Lowry (1971) we have converted
their values of the aftershock rate to equivalent values for all shocks with magnitude
3 1 - 0 using a b-value (equation (I)) of 1 .O. This value of b is mid-way between
the two values they present (Table 4). The recomputed rates are shown in Fig. 3
together with the value we obtain and also two rates for the latter part of 1968 during
which the New Zealand Seismological Observatory located a sufficient number of
aftershocks for a rate to be determined. It can be seen that over a long period of
time the rate of aftershock occurrence can still be represented by an equation of the
form of (2) with constant p . Indeed, the value of p found by least-squares for this
complete data set is 1 *06+0.05, nearly identical to that found by Adams & Lowry
(1971), so that the rate during our survey could have been quite accurately predicted.
Distribution of hypocentres
In some aftershock sequences, precisely determined hypocentres appear closely
to define a planar surface, thought to represent the fault plane of the main shock,
although the distribution of events along that surface may be complex in detail
171.75"
7
172"E
FIG.4. Map of seismograph stations and epicentres of the late aftershocks. Stations
are indicated by large solid triangles with numbers corresponding to Table 1.
Solid circles represent class A epicentres; larger open circles, class B epicentres;
smaller open circles, class C epicentres. The solid diamond on the trace of the
Glasgow Fault is the epicentre of the main shock.
43
Long-term behaviour of an aftershock sequence
1
2or-
q
5 --
10
F
-
20
l
5
I
I
__--1.---- - ._ _ _
FIG. 5 . Cross-sectionsof the late aftershock activity as indicated in Fig. 4. Only
class A and B locations are shown. Horizontal and vertical scales are equal.
~
'r
0
I
-
I
I
-
-
i
-
2
6
I
i-
a
10
w
0
12
14
10
18
20
15
20
25
30
OF E V E N T S
FIG. 6. Distribution with depth of class A and B locations.
0
5
10
NUMBER
44
R. Robinson, W. J. Arabasz and F. F. Evison
(e.g. Eaton, O'Neill & Murdock 1970). In many cases, however, the aftershock
hypocentres appear to occupy a volume and define no single fault plane (e.g.
Hamilton 1972).
Adams & Lowry (1971) found that epicentres of aftershocks occurring during the
first 40 days of the Inangahua sequence occupied an elliptical area about 45 km by
25 km extending south-south-west from near the epicentre of the main shock. The
distribution of focal depths is not well known; calculated depths were all near 12 km
but control was not good. Epicentres for 155 micro-aftershocks located i n this study
(late aftershocks) are shown in Fig. 4. Another 50 class D locations that were not
plotted have a distribution similar to that of the better quality locations. The late
aftershocks define an elongate ellipse, 40 km by 15 km, centred on the Glasgow Fault
and extending south-south-west from near the epicentre of the main shock-a
distribution quite similar to that of the early aftershocks although narrower in width.
(We note, however, the different magnitude ranges considered.) Cross-sections
(Fig. 5) indicate that no planar surfaces are defined by the late aftershock hypocentres,
although their density appears to be greatest directly beneath the surface trace of the
Glasgow Fault. The aftershock distribution as a function of depth is shown in Fig. 6.
Focal mechanisms
The nature of the regional tectonic stress at Inangahua is well known. Surface
faulting accompanying the Inangahua earthquake was dominantly thrusting (with the
east side upthrown) along with a smaller left-lateral component; the faulting that
occurred in 1929 on the White Creek Fault was of similar type, though with much
greater displacement. This surface evidence has been confirmed by first motion
studies. Adams & Lowry (1971) have plotted first motions for the main Inangahua
shock and have summarized those for the early aftershocks; they conclude that thrust
faulting with a northerly strike occurred in most shocks including the main one, and
they estimate the fault plane to dip eastwards at about 45". Results publishcd by
Johnson & Molnar (1972) for the main shock are consistent with this.
Further confirmation is given by first motions of some well-recorded early aftershocks, of which the locations have been recomputed having particular regard to
focal depth. These shocks were well distributed through the aftershock volume.
The first motions are shown as a composite in Fig. 7(a), which also shows the first
motions of the main shock replotted from Adams & Lowry (1971). The nodal plane
that strikes N 19" E and dips 44" E implies thrust faulting with a small left-lateral
component.
A very different type of faulting is indicated by a composite focal mechanism for
the late aftkshocks. A composite plot of the first motions from all well-located shocks
(class A and B locations) is shown in Fig. 7(b). These shocks were also well distributed
through the aftershock volume. Again one of the nodal planes strikes close to the
surface trace of the Glasgow Fault. Choosing this as the most likely fault plane
one finds that normal faulting was dominant, on a plane dipping 78" NW, and that
again there was a small left-lateral component.
Thus, between the early aftershocks occurring within 40 days of the main event
and those recorded 3.6 years later, the main fault movement has changed from thrust
to normal and the dip of the fault planes has swung through about 60". Since
the internal consistency of each first motion plot is rather high it appears that at each
stage the seismogenic conditions within the aftershock zone were to a rather high
degree uniform.
Discussion
Considering the results we have presented, the long-term behaviour of the
Long-term behaviour of an aftershock sequence
N
N
8
N 19'E
6 =N 2 7 O E
FIG. 7. Composite focal mechanism diagrams on equal-area stereographic
projections of the upper focal hemisphere. Solid circles are compressions; open
circles, dilatations. Each nodal plane is specified by a strike 9 and a dip 6.
Idealised axes of compression and tension are indicated by P and T respectively:
(a) Main shock (large circles) and early aftershocks (small circles); (b) Late
aftershocks.
45
46
R. Robinson, W. J. Arabasz and F. F. Evison
Inangahua aftershock sequence can be characterized as follows:
(1) The variation of frequency of occurrence with magnitude is effectively constant
with the value of the constant b in the Gutenberg-Richter relation (1) near unity;
(2) The rate of aftershock activity as a function of time can be represented by a
relation of the form n(t) = At-''05 over the 3.6-yr period of observation;
(3) The elliptical distribution of aftershock epicentres remained roughly constant,
although there is an indication of narrowing about the trace of the Glasgow Fault
during the late aftershocks (note, however, that the distribution of small, ML < 3.8
early aftershocks is not known);
(4) The focal mechanism of the aftershocks has changed radically from the initial
thrusting, similar to the mechanism of the main shock, to normal faulting; a small
left-lateral component remains throughout the aftershock sequence, however.
The first three properties of the Inangahua aftershock sequence are essentially
what would be expected from a simple extrapolation of the short-term behaviour
and can be explained, with varying degrees of success, by several proposed models of
the aftershock process (e.g. Scholz 1968; Dieterich 1972; Burridge & Knopoff 1967).
The change in focal mcchanism, however, is unexpected and provides, perhaps, the best
clue to a physical understanding of the Inangahua sequence.
The mechanism of the late aftershocks indicates normal faulting in a region where
the tectonic stress is known to be compressive. It is difficult to see how pervasive
normal faulting can set in, apart from recovery after overshoot, unless the tectonic
stress is reduced to a low level and also material is somehow removed at depth. An
earthquake and its aftershocks serve, of course, to reduce the tectonic stress and there
is no great difficulty in supposing a low compressive stress level to have been reached
throughout the source region at the time of the late aftershocks. As for the removal
of material at depth in the period following an earthquake, this may be a consequence
of pre-earth quake d i 1atancy .
On the dilatancy hypothesis (Nur 1972; Scholz, Sykes & Aggarwal 1973) there
is an inflow of pore fluids to the source region before an earthquake. Following
the earthquake we can visualize that the stress level will be reduced below that necessary
to cause dilatancy. Thus, the newly created pore spaces will tend to close, being held
open, however, until the enclosed pore fluid, under very high pressure, can flow out
of the source region again. In the broadest terms the observed normal faulting in the
late aftershocks can be attributed to subsidence following such an outflow of fluid.
Evison, Robinson & Arabasz (1973) have shown, schematically, how such a process
can explain the observed orientation of the fault plane for the late aftershocks in the
Inangahua sequence.
The change in focal mechanism between the early and late aftershocks may seem
to be in conflict with the apparent constancy of the rate of decay of aftershock activity
which would seem to imply a constancy in the physical process causing the aftershocks.
Aftershocks will occur, however, as long as the applied shear stress exceeds the
strength of a potential fault plane, whatever the cause of that stress field or direction
of that fault plane. Although the nature of the stress field at Inangahua has changed
substantially, the very high pore pressures within the entire aftershock zone will cause
any potential fault to be very weak as the effective normal stress is reduced (Hubbert &
Rubey 1959; Healy et al. 1968). Thus the rate of aftershock activity will be governed
by the level of the pore pressure as long as sufficient stresses are present (Rayleigh,
Healy & Bredehoeft, 1972). As the high pore pressures decay due to outward flow
of fluid the aftershock rate will also decay, any potential fault becoming stronger as
time goes on. Detailed calculations would be necessary to examine the exact nature
of such a process and to see if it could provide an explanation of the observed decay
rate. (This process does not necessarily preclude a variation in the aftershock rate
from that predicted by (2) if the stresses had at some time dropped to levels lower
than the strength of the aftershock region sometime between 1969 and 1972.) Nur
Long-term behaiiour of an aftershock sequence
47
& Booker (1972) and Booker (1974) have proposed that, near the ends of a just
ruptured fault, pore fluids will flow from regions of relative compression to regions
of relative dilation and serve to trigger aftershocks by increasing the pore pressure
in the latter regions. Their calculations also show that, as a result, the fault will be
slowly reloaded, perhaps causing additional aftershocks. Such processes could be
superimposed on that described here.
Finally, we note that the change in focal mechanism during the Tnangahua aftershock sequence shows that the stress orientation indicated by micro-earthquakes
occurring over a substantial area may be very different from that of the regional
tectonic stress. As late aftershocks for large shallow earthquakes continue for years
and may possibly be confused with the normal background microseismicity, it would
be prudent to examine carefully the nature of activity being observed during microearthquake surveys before making conclusions based on the focal mechanisms.
Acknowledgments
Mr. M. E. Reyners assisted with the field work. We thank Dr. R. D. Adams for
comments and suggestions. Two authors (Robinson and Arabasz) gratefully acknowledge the support of post-doctoral fellowships at their respective institutions. The
study was made possible by an equipment grant from the University Grants Committee, Wellington.
Institute of Geophysics,
Victoria University of Wellington,
Wellington.
W. J. Arabasz:
Geophysics Division,
Department of Scientific and Industrial Research,
Wellington.
References
Adams, R. D., Eiby, G. A. & Lowry, M. A., 1968. The Inangahua Earthquake,
Preliminary Seismological Report, N.Z. Dept. Sci. Ind. Res. Bull., 193, 7-16.
Adams, R. D. & Lowry, M. A., 1971. The Inangahua Earthquake Sequence, 1968,
N.Z. R.SOC.Bull., 9, 129-135.
Booker, J. R., 1974. Time Dependent Strain Following Faulting of a Porous
Medium, J . geophys. Res., 79, 2037-2044.
Brune, J. N. & Allen, C., 1967. A Microearthquake Survey of the San Andreas
Fault System in Southern California, Bull. seism. SOC.Am., 57, 277-296.
Burridge, R. & Knopoff, L., 1967. Model and theoretical seismicity, Bull. seism.
SOC.Amer., 57, 341-371.
Dieterich, J. H., 1972. Time dependent friction as a possible mechanism for aftershocks, J . geophys. Res., 77,3371-3781.
Eaton, J. P., O’Neill, M. & Murdock, J. N., 1970. Aftershocks of the 1966 ParkfieldCholame, California, Earthquake: a Detailed Study, Bull. seism. SOC.Am., 60,
1151-1 197.
Eiby, G . A., 1971. Seismic Regions of New Zealand, R. SOC.N.Z. Bull., 9, 153-160.
Evison, F. F., Robinson, R. & Arabasz, W. J., 1973. Late Aftershocks, Tectonic
Stress, and Dilatancy, Nature, 246,47 1-473.
Geiger, L., 1912. Probability Method for the Determination of Earthquake
Epicenters from the Arrival Time only (Trans. of 1910 German article)., Bull.
Sr Louis University, 8, 56-7 1.
48
R. Robinson, W. J. Arabasz and F. F. Evison
Gibowicz, S. J., 1963. Magnitudes and Energy of Subterane Shocks in Upper
Silesia, Studia Geophys. Geod., 7, 1-18.
Hamilton, R. M., 1966. The Fiordland Earthquake Sequence of 1960 and Seismic
Velocities beneath New Zealand, N.Z. J. Geol. Geophys., 9, 224-238.
Hamilton, R. M., 1972. Aftershocks of the Borrego Mountain Earthquake from
April 12 to June 12, 1968, The Borrego Mountain Earthquake of April 6, 1968,
U.S.G.S. Prof. Paper 787, 31-54.
Healy, J. H., Rubey, W. W., Griggs, D. T., & Raleigh, C. B., 1968. The Denver
Earthquakes, Science, 161, 1301-1310.
Hubbert, M. K., & Rubey, W. W., 1959. Role of Fluid Pressure in Mechanics of
Overthrust Faulting, Bull. Seism. SOC.Am., 70, 115-166.
James, D. E., Sacks, I. S., L a o , E. & Aparicio, P., 1969. On locating Local Earthquakes Using Small Networks, Bull. seism. SOC.Am., 59, 1201-1212.
Johnson, T. & Molnar, P., 1972. Focal Mechanisms and Plate Tectonics of the
South-West Pacific, J. geophys. Res., 77,5000-5032.
Lee, W. H. K., Bennett, R. E. & Meagher, K. L., 1972. A method of estimating
magnitudes of local earthquakes from signal duration, U.S. Geological Survey,
Open File Report.
Lee, W. H. K. & Lahr, J. C., 1972. HYP071: A Computer Program for determining
the hypocenter, magnitude, and first motion pattern of local earthquakes, U.S.
Geological Survey, Open File Report.
Lensen, G. J. & Otway, P. M., 1971. Earthshift and Post-Earthshift Deformation
Associated with the May, 1968 Inangahua Earthquake, New Zealand, R . SOC.
N.Z. Bull., 9, 107-116.
Lensen, G. J. & Suggate, R. P., 1968. The Inangahua Earthquake-Preliminary
Account of the Geology, N.Z. Dept. Sci. Ind. Res. Bull., 193, 17-36.
Nur, A., 1972. Dilatancy, Pore Fluids, and Premonitory Variations of t,/t, Travel
Times, Bull. seism. SOC.Am., 62, 1217-1222.
Nur, A. & Booker, J. R., 1972. Aftershocks caused by pore fluid flow?, Science,
175, 885-887.
Page, R., 1968. Aftershocks and microaftershocks of the Great Alaska Earthquake
of 1964, Bull. seism. SOC.Am., 58, 1131-1168.
Raleigh, C. B., Healy, J. H. & Bredehoeft, J. D., 1972. Faulting and Crustal Stress at
Rangely, Colorado, in Geophysical Monograph 16, American Geophysical
Union, Washington, D.C.
Ranelli, G., 1969. A Statistical Study of Aftershock Sequences, Ann. Geojk, 22,
359-397.
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Am., 58,1117-1 130.
Scholz, C. H., Sykes, L. R. & Aggarwal, Y. P., 1973. The physical basis for earthquake prediction, Science, 181, 803-810.
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Appendix
Determination of micro-earthquake magnitude
The basic form of any amplitude-magnitude relation is
M = log A -log A0
where A is one of several possible measures of amplitude and A,(A) is the distance
dependent amplitude of a magnitude zero event. Eaton et al. (1970) discuss the many
problems associated with such a magnitude scale when applied to micro-earthquakes.
Long-term behaviour of an aftershock sequence
49
Brune & Allen (1967) have developed the most commonly used micro-earthquake
magnitude relation of this general form.
Considering possible variations in the &(A) factor (due to different instrument
responses and geologic environment) we have developed our own amplitude magnitude relation
M = logA+alogR+b
(2)
where A is the zero to peak amplitude (mm) of either the P or S phase (there will be
separate formulas for the two phases) corrected to a gain of 1.6 x lo6 (30 db attenuation on our instruments), R is the radial distance from the hypocentre to the station
(km), and a and b are constants. The constants a and b were determined by the
method of Gibowicz (1963) using only local micro-earthquakes. The results are
M p = log Ap+2-75 log R - 3 . 4 4
for P waves
(3)
M, = log As+2.71 log R - 3 . 2 8 for S waves
(4)
These formulas are close to those that would have been derived from the Brune &
Allen magnitude formula. The values of the constant a imply an amplitude decay
proportional to R - 2 ’ 1 .
In order to overcome the problem involved with a small range in observable
amplitude (our instruments clip all amplitudes > 20mm) we have also derived a
magnitude scale based on event duration. Lee et al. (1972) found a scale of the form
M , = ~ + logz+cA
b
(5)
where A is the epicentral distance, z is the signal duration, and a, b, and c are constants,
that accurately determines micro-earthquake magnitudes in central California. We
obtain, by a least-squares method, a similar formula from our data:
M , = -1.51+1*7410g~+0.019R.
(6)
Magnitudes determined from signal duration are less dependent on focal
mechanism or local station effects than those determined from amplitudes, especially
if a small number of observations are available. The magnitude of an event can be
determined from the data at only one station if the S - P interval for that event can be
estimated since R depends directly on that interval.
The standard deviation of both M , and
= (M, +M J / 2 from the local magnitude
as determined by Wood-Anderson or Wilmore seismographs is of the order of 0.3.
D