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Geophys. J. R. astr. SOC. (1977) 50,381-394
Fault plane solutions using relative amplitudes of P and pP
* R. G. PearCe School of Physics, The University,
Newcastle-on-Tyne NEI 7RU
Department of Geophysics and Planetary Physics,
Received 1976 December 22; in original form 1976 October 22
Summary. One way of finding the fault plane orientations of
small shallow earthquakes is by the generation of theoretical
P-wave seismograms to match those observed at several distant
stations. Here, a technique for determining the uniqueness of fault
plane solutions computed using the modelling method of Douglas et
al. is described. Relative amplitudes of P and pP, and their
polarities if unambiguous, are measured on the observed seismograms
to be modelled, and appropriate confidence limits are assigned to
each measure- ment. A systematic search is then made for all fault
plane orientations which satisfy these observations.
Examples show that if P and pP are not severely contaminated by
other arrivals, a well-defined and unique fault plane orientation
can often be com- puted using as few as three stations well
distributed in azimuth. Further, even if pP is not identifiable on
a particular seismogram, then an upper bound on its amplitude -
deduced from the observed coda - still places a significantly
greater constraint on the fault plane orientation than would be
provided by a P onset polarity alone. Modelling takes account of
all such information, and is able to further eliminate incompatible
solutions (e.g. by the correct simu- lation of sP). It follows that
if solutions can be found which satisfy many observed seismograms,
this places high significance on the validity of the assumed
double-couple source mechanism.
This relative amplitude technique is contrasted with the
familiar first motion method of fault plane determination which
requires many polarity readings, whose reliabilities are difficult
to quantify. It is also shown that fault plane orientations can be
determined for earthquakes below the magni- tude at which first
motion solutions become unreliable or impossible.
1 Introduction
The modelling method of Douglas et al. (Hudson 1969a, b;
Douglas, Hudson & Blamey 1972) can be used to compute
theoretical long-range P-wave seismograms for shallow earth-
quakes. The method assumes a doublecouple source mechanism over an
extended fault *Present address: MOD(PE), Blacknest, Brimpton,
Reading, Berks RG7 4RS.
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382 R. G. Pearce plane of the type described by Savage (1966),
and allows for the effects of a horizontal plane layered velocity
structure at both the source and receiver. Account is also taken of
anelastic attenuation, geometrical spreading and the response of
the recording instrument. The aim is to vary the model empirically
until the closest possible match is obtained between theoretical
and observed seismograms at several recording stations
simultaneously. In parti- cular, a fault plane solution is obtained
by searching for a source orientation which repro- duces the
polarities and relative amplitudes of direct P and the free surface
reflections pP and sP as observed on the seismograms.
Using this method, Douglas et al. (1974) were able to find a
fault plane orientation which yielded theore tical P-wave
seismograms closely resembling those recorded at three short period
arrays from an earthquake in East Kazakhstan. Similarly, Cullen
& Douglas (1975) modelled three earthquakes in Southeast
Europe. However, because these models were determined by a process
of informed trial and error, no measure of the confidence limits,
or of the uniqueness of the resulting fault plane orientations
could be given. Such a measure is clearly necessary in order to
assess the value of fault plane Orientations deduced by model-
ling; the present paper a i m s to provide this.
2 Method
For a shallow earthquake, we expect P and pP to suffer the same
fractional loss of amplitude along their paths, except near the
source, where pP may encounter seismic discontinuities or
scattering centres above the focus which are not traversed by P. If
the earthquake is small (mb < 5.6) the source pulse is likely to
last for less than one second, so that P and pP should be simple
and well separated on a short-period seismogram. In such cases we
expect the main signatures of the fault plane orientation on the
seismogram to be the polarities and relative amplitudes of P, pP
and sP. Therefore, provided an allowance can be made for any near-
source differential effects between the direct and surface
reflected phases, the range of fault plane orientations compatible
with a particular seismogram can be computed.
The possibility of utilizing the information contained in
surface reflections in order to constrain fault plane orientations
has also been suggested by Langston & Helmberger (1975). Honda
& It8 (1951) and Kasahara (1963) compared observed and
predicted amplitudes of principal phases from deep focus
earthquakes, assuming previously computed fnst motion solutions.
They concluded that the doublecouple mechanism was a reasonable
model, though there was substantial variability in the consistency
of their observations. Further, these authors did not use amplitude
information to determine fault plane orientations, and no
quantitative analysis was attempted. Randall & Knopoff (1970)
tested the observed distribution of absolute P-wave amplitudes on
the focal sphere against several source models. The difficulty of
correcting each observation for the effects of the path and
receiver regions - in p?rticular anelastic attenuation - restricts
such studies to long-period seismograms, which in turn precludes
the use of shallow earthquakes.
In the method introduced here a systematic search is made for
all fault plane orientations which are compatible with a series of
short-period P and pP observations from shallow earth- quakes. By
using relative amplitudes, the need to correct for anelasticity
along the path is avoided, so permitting the use of short-period
seismograms, in which P and pP are separated even for intracrustal
earthquakes. Array stations are employed in order to improve signal
to noise ratio, and because they are sited on simple, well
determined crustal structures. Any single seismometer stations
could also be used, remembering the preference towards good
coverage of the focal sphere. The computational procedure is now
described.
Let the seismic radiation from a point source be defined in
spherical polar coordinates
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Fault plane solutions 383
Fwre 1. (a) Coordinate system used to define seismic source
mechanism. (b) Double couple about the X, axis. (c) Polar diagram
of far field P-wave radiation resulting from double couple source
shown in (b).
(I, 8,#) about the source, where 8 = 0 is coincident with the X,
Cartesian axis and 4 is measured in a right-handed sense from the
X, axis - Fig. l(a). Then for a double-couple force system acting
about the X, axis as shown in Fig. l(b), the time-dependent part of
the far-field P-wave amplitude at unit distance from the source is
given by (see, e.g. Honda 1957)
sin 28 COSG i A, (e; $1 = - - - 4n up A + 2 p K 1 1
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3 84 R. G. Pearce where X and I.( are L a m B s parameters, up
is the P-wave velocity in the source medium, K is a constant
dependent upon the magnitude of the couples, and P denotes the unit
vettor. Similarly the S-wave amplitude is given by
K 1 1
4n us cc As@,@)=-- - - ( c o s ~ ~ cosdB-cosesin@+)
P Figure 2. (a) Definition of source orientation in space. (b)
Definition of P-wave azimuth and take-off angle from the
source.
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Fault plane solutions 385
where us is the S-wave velocity in the source medium. The
integrated effect over an extended source has little influence on
the radiation pattern for the small faults discussed here, and will
be neglected. It is clear from equations (1) and (2) that, although
the ratio of P and pP amplitudes at the focal sphere is independent
of all geophysical parameters, this is not true for P and sP. For
simplicity, consideration is initially limited to P and pP.
Fig. l(c) is a polar diagram of the P-wave radiation Ap ( O , #
) ; we note the familiar degeneracy under interchange of the fault
plane (arbitrarily chosen as the plane X2X3) and the auxiliary
plane, X1X2. The source orientation is determined by the
orientation of th is coordinate system in space, which is here
defined by the dip 6 of the fault plane, the slip angle $ in this
plane, and its strike u from north, as shown in Fig. 2(a). The
direction of the P-wave emergence to a given station is defined by
its azimuth E , and its angle of take-off (Y - Fig. 2(b).
To calculate the amplitude of P at source for a particular fault
plane orientation, we require the direction vector x = { xl, x 2 ,
x3j of the P ray leaving the source expressed in the X coordinate
system. This can be found by applying a series of rotations to the
direction vector along X3, namely {O,O, l}. The required direction
vector is given by
where q = [ - u is the azimuth of the recording station from the
strike, and p = + 1 for P and - 1 for pP. The amplitude is then
given by
K 1 1 A , = - - - sin (2 cos-' (x3)) cos (tan-' (XZ/XI))f.
4n up X + 2 p (4)
A computer program has been written to search systematically for
all source orientations which are compatible with an acceptable
range of P and pP amplitude ratios, deduced from an observed
seismogram. Differential loss of amplitude of pP above the source
is allowed for by a multiplicative factor, the calculation of which
is discussed later. For each station 4' and (Y are known, and ($,
6, q) space is searched using a mesh size d (normally 5") within
the bounds d < $ 6 n; d < 6 < n and d < q < 2n. This
choice of bounds avoids duplication of orientations while
maintaining the interchange of fault and auxiliary planes as
separate solu- tions. The procedure can be repeated for
observations at other stations; orientations which are acceptable
at each station are transformed into the fixed system ($,ti, u) and
only those acceptable at all stations are retained. The notion of
dividing orientation space into accept- able and unacceptable
regions is used in preference to the maximization of a criterion
function, because the aim is+o establish the range of solutions
which is compatible with the data. Anomalous observations which
require explanation are then emphasized.
A clear representation of all acceptable orientations ( $ i , S
i , ui) is provided by plotting each as a unit vector drawn at an
angle ui from the Cartesian point ( $ i , S i ) as shown in Fig. 3.
Different combinations of I) and 6 represent various types of
fault, and these are indi- cated by lower hemisphere stereographic
projections superposed on this plot. The projec- tions are oriented
for northerly strike (u = 360" - upward vector); other strike
directions are interpreted by equivalent rotations. The positions
of some important types of fault on the plot are shown in Fig. 3 -
there is no need to consider the effect of interchanging fault and
auxiliary planes, as the resulting fault types are represented
elsewhere on the plot.
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386 R. C. Pearce
slip angle in fault plane CJ 5" 45" 90" 135" 180" . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + + +
+ + + + + + + + ?eO\-hy",0ntp' p.p +[vyti:al+dil: $!PI+ + + + + + +
+ + + + + + + + st r ,ke+ + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + q (r' . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . ................................
a + + + + + + + c + + + + + + + %445"+ + + + + + +
+ t i + + + + - 3 + + + + + + + (0
+ + + + + + + + Lc + + + + + + +
O + + + + + + + 4 + + + , + + + + go".""h. + + + + +
lev? + + + +[si+nisty[ + + + shear
near-vertical
:trip $p+ + + + + + + + + + +
+ + + + + + +
+ + +
:@ + + + + + + + + + + + + + + + + + +
+ + + + + +
+ t
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + +
+ + + + + + + + + + + + + t
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + +
+ + +
:@ + + + + + + + + + + + + + + + + + +
+ + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + ::e + + + + + + + + + + + + + + + + + + + + + + + +
+
vertical strike slit
sinistral [dytyl],
+ + + +
shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .
135". + + + + + + . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
180. + + + + + + + + + + + + + + + + + +
Figure 3. Method of representing acceptable fault plane
orientations in terms of slip direction $ , dip 6 and strike u, as
defiied in Fig. 2(a). Acceptable orientations are plotted as
vectors from the Cartesian point defining $ and 6 , in the
direction of the strike u. Lower hemisphere stereographic
projections indicate the type of fault plane orientation
represented by various combinations of $ and 6 , and are shown
oriented for strike u = 360" (northerly). In each case the fault
plane is shown by a thick line; the auxiliary plane by a thin line.
Shaded quadrants are negative. Different parts of the plot
characterize various fault types, and some of these are shown.
Where the interchange of fault and auxiliary planes yields a
different fault type, this is shown in square brackets.
3 Practical considerations
Before applying the relative amplitude method, a number of
practical points must be dis- cussed. Identification of pP is
achieved by looking for a prominent phase following P, which is
common to several seismograms - focal depths read from bulletins
are not sufficiently reliable for this purpose. Where no clear pP-P
time can be deduced for an earthquake, obser- vation of the maximum
amplitude in the P-wave codas on each seismogram can be used to set
an upper bound to the pP amplitude.
The method of specifying the pP/P amplitude ratio, and its
reliability, was chosen to be compatible with the measurements that
can be made from a seismogram. The possible ranges of both P and pP
amplitudes are defined separately in arbitrary amplitude units. In
addition, the polarities of these phases are each defined as
positive, negative or uncertain - implying the inclusion of one or
both senses for each amplitude range specified. Thus, for
uncertain
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Fault plane solutions 387
polarity observations, both a positive and negative amplitude
range will be acceptable, and for small or 'nodal' observations, a
lower bound of zero with an undefined polarity will cause a range
of amplitudes about zero to be acceptable.
The need to specify a finite range of amplitudes arises from
observational uncertainty caused by noise, by interfering arrivals,
and from the need to allow for any differences in the frequency
content of P and pP, which will affect their relative amplitudes as
recorded by a short-period instrument. The aim is to specify 100
per cent confidence limits on the ampli- tudes, which will
therefore usually be wide (typically spanning at least a factor of
two for pP observations) but which will directly represent the
observational reliability of each read- ing. Pearce & Barley
(1977) have introduced a method of adding synthetic noise to
theoreti- cal seismograms in order to establish the amount of
signal distortion that is attributable to noise on any given
observed seismogram.
Changes in the relative amplitudes of P and pP between the
source and receiver will, for shallow earthquakes, be limited to
effects on pP above the source - the most likely causes being
scattering, attenuation, and energy partitioning at seismic
discontinuities other than the free surface. Where modelling (or
independent evidence) provides a determination of the velocity
depth structure, a correction can be made for the energy
partitioning by computing the Zoppritz (1919) equations at
successive discontinuities from the source layer to the sur- face,
and back to the source layer. For intracrustal earthquakes which
yield simple P-wave seismograms consisting primarily of P, pP and
sP, calculation shows that such energy losses are small, provided
there is no sea layer, and are neglected.
4 Application of the method
Two examples are now presented as a practical test of the
relative amplitude method, both as a means of assessing the
uniqueness of fault plane orientations already deduced by model-
ling, and as a means of computing fault plane solutions
directly.
Figs 4(a), (d) and (g) show the P-wave seismograms observed
teleseismically at the short period arrays YKA, WRA and GBA
respectively, from an earthquake in East Kazakhstan (Earthquake 1
of Table 1). Douglas et al. (1974) used the computational method of
Hudson (1969a,b) and Douglas et al. (1972) to generate
corresponding theoretical seismograms assuming a focal depth of
25.4 km. This choice of depth enabled the second arrival at YKA and
WRA (whose amplitude is too small to be measured at GBA) to be
modelled aspP. By assuming dip slip on a fault dipping at SO"
towards YKA, Douglas et al. (1974) were able to closely match the
polarities, relative amplitudes and pulse shapes of P and pP at the
three stations. Details of their model (designated Model 1) are
given in Table 2, and their theoreti- cal seismograms are
reproduced in Figs 4(b), (e) and (h). The closeiiess of match
provides adequate confirmation that pP has been correctly
identified.
The question is now asked: 'what range of alternative fault
plane orientations would yield theoretical seismograms whose P and
pP pulses would also match the observed seismograms?'
Table 1. Earthquakes used in this paper (USCGS parameters).
Date Origin time Region Location m b h m s
1 1969May1 04 00 08.7 East Kazakhstan Lat. 43.98' N 4.9
2 1972 January 12 08 15 46.1 GulfofSuez Lat. 27.53' N 5 .1
Long. 77.86'E
Long. 33.75" E
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388 R. G. Pearce
P:6.0 to 8.0;we pP: 4.0 to ll.O;-Ve (a) observed
at Y K A c -6.0' A-73.3'
for YKA
I
(cIModel 2 computed for YKA
( 1 1 Model 2
for GBA computed I
P: 8.0 to lL.O;+ve
-
computed for WRA
I
P:l5 to 6.0:+vel-ve SR
A-30.3" Y
h I;
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Fault plane solutions 389 Table 2. Models used for East
Kazakhstan earthquake (Model 1 taken from Douglas ef al. (1
974)).
(a) Source parameters
Focal depth = 25.4 km Radius of fault plane = 1.25 km (circular)
Stress drop across fault = 100 bar Fracture velocity = 0.6 us,
where us = source layer S-wave velocity
Dip of fault plane 6 = 130" 90" Direction of slip in the fault
plane J, = 90" 180"
Model 1 Model 2
Strike of fault plane u = 96" 101"
(b) Source structure
P-wave velocity S-wave velocity Density Thickness kmls km/s g
cm-3 km
Layer 1 4.80 - 2.7 1 .o Layer 2 6.15 3.5 2.8 30.6 Layer 3 8.2 -
3.3 00
Where S wave velocity (us) is not listed us = up/J3 where up is
the P-wave velocity.
(c) Receiver structures
YKA - Hasegawa (1971); WRA - Underwood (1967);CBA - Arora
(1969).
Realistic bounds on the possible amplitudes and polarities of
these two phases are specified in Fig. 4, in the manner developed
above. Because of the simplicity of the observed seismo- grams, any
loss of pP energy above the source is assumed to be negllgible. The
acceptable fraction of orientation space acceptable at each station
is shown in Table 3, and Fig. 5 shows a plot of all fault plane
orientations compatible with these three observations, displayed as
described in Fig. 3. These acceptable solutions occupy only 1.7 per
cent of orientation space - which shows that as few as three
seismograms can contain enough information to severely constrain
the orientation of an assumed double-couple.
Nearly all the acceptable orientations are clustered around that
used by Douglas et al. (1974) (shown with a single arrowhead
together with its 'degenerate partner' resulting from interchange
of fault and auxiliary planes). This cluster centres on the reverse
thrust 45' dip slip type of fault (see Fig. 3), and some features
familiar to any first motion worker can be clearly seen. While the
dip is constrained to within +15' and the slip direction to within
a maximum of ? 45", the strike is very poorly defined - as the
angular spreads of up to 180" in Fig. 5 indicate. When observed in
first motion solutions of dip slip faults, this effect is caused by
the absence of any nodal plane passing through the teleseismic
annulus on the focal sphere, within which observations are
available. A more general case of the same phenomenon occurs here:
that is, there are no large P-wave amplitude differences within
this
Figure 4. P-wave seismograms from East Kazakhstan earthquake,
1969. (a), (d) and (8) are observed at Yellowknife (YKA),
Warramunga, (WRA) and Gauribidanur (CBA) respectively; azimuth and
epicentral distance A to each station are shown. Bounds on
acceptable amplitudes and polarities of P and pP as used in the
fault plane search are also indicated, with amplitudes in arbitrary
units. (b), (e) and 01) show corresponding theoretical seismograms
computed using the model of Douglas ef al. (1974) (Model 1). (c),
(0 and (i) are corresponding theoretical seismograms computed using
Model 2. Note that, although Model 2 generates a good match for P
and pP at all three stations, its predictions of sP are in poor
agree- ment. 1* (the ratio of travel time to anelastic quality
factor Q) is assumed to be 0.2 s for each seismic path.
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390 R. G. Pearce Table 3 . Constraint on fault plane orientation
imposed by relative amplitudes of P and pP.
Station Fraction of orientation space acceptable
East Kazakhstan earthquake Gulf of Suez earthquake (%) (%I
16.0 EKA - GBA 50.0 5.4 WRA 10.2 - YKA 8.1 6.1 At three stations
1.7 0.3
annulus - or within the equivalent upper hemisphere annulus,
which is an additional part of the focal sphere sampled by this
method.
A small group of vertical strike slip solutions are also
acceptable (Fig. 5 ) highlighting the importance of testing models
for uniqueness. While empirically perturbing the orientation of the
dip slip model to achieve the closest match, the possibility of the
fundamentally differ- ent dynamic processes implied by a vertical
strike slip fault might be entirely overlooked, although such a
model may fit the observations.
An additional constraint can be placed on the fault plane
orientation using the relative amplitudes of P and 9. Douglas et
al. (1974) noted that, using their model, the second arrival
observed at GBA - Fig. 4(g) - was reproduced closely as sP. This is
clearly not true of all orientations shown in Fig. 5 . For example,
the solution designated Model 2 in Table 2 (which is shown with a
double arrowhead in Fig. 5 ) incorrectly generates the sP observa-
tions, while the P and p P pulses are, of course, still compatible
- see Fig. 4(c), (f) and (i). This testifies to the importance of
sP in fault plane determination.
slip angle in fault plane C,J 5" 30" 60" 90" 120" 150" 180" . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . g0"e*++ + + t + + + + + + + + + + + + + + + + + + + + + + +
+ + +
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . + + + + + + + + b ~ ~ ~ \ L L ~ L L L + 1 + + + + + + + + +
+ + + + + + + +
+ + + + + +
a + + + + + + + - + + + + t + + t t t t t t + + + + + + + + + +
+ + + + + 73 + + ~ + + + + + + + , + + + + + + + + C + + + t + + +
+ + + + + + + + +
+ + + \ ,, + + + + + + t + + + + + * + i + + + + + + t t t + + t
+
.- ++++++++++++++++++++++++++*+++*,+++++++++
+ + + + + + + + + + + + t + + + + + + + + + + + + + t t + +
+
1 8 O y + + + + + + + + + + + + + + i + + + + + + t + + t + + +
+ + + + t t + + Figure 5. East Kazakhstan earthquake 1969. Fault
plane orientations which are compatible with the polarities and
relative amplitudes of P and p P in all three observed seismograms,
as specified in Fig. 4. Note the separate group of vertical strike
slip solutions at extreme left and right edges of plot. Single and
double arrows show orientations used for Models 1 and 2
respectively (accurate to the nearest increment in the mesh).
Because either nodal plane may be the fault plane, each orientation
appears twice.
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Fault plane solutions 39 1
(a) observed at YKA t = 345.9O A= 86.6'
Ib) correlogram at YKA
k) computed for YKA t * = 0 . 6 ~
Id) observed at GBA t 199.80 A= 43.0'
(el correlogmm at GBA
It) computed tor GBA t'= 0.7s
Ig) observed at EKA
C = 326.5O A = 38.5O
(hl correlogram at EKA
I iI computed for EKA t *= 0.5s
t2 P 15to20,+ve
P: 0.9 to 1.1: -ve
Figure 6. P-wave seismograms from Gulf of Suez earthquake 1972.
(a), (d) and (g) are observed at YKA, GBA and Eskdalemuir (EKA)
respectively, annotated as in Fig. 4 . @), (e) and 01) are the
corresponding correlograms, indicating the arrival of energy with
the correct azimuth and phase velocity. (c), (0 and (i) are
corresponding theoretical seismograms, computed using the model of
Table 4.
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392 R. G. Pearce Apart from providing a clear assessment of the
uniqueness of a fault plane solution com-
puted by modelling, the relative amplitude method offers a means
of computing fault plane orientations directly. Moreover, in
another example it is shown to be effective in doing t h i s using
seismograms with low signal-to-noise ratio, where first motion
observations are unreliable or impossible.
Fig. 6(a), (d) and (g) show observed seismograms at the three
arrays YKA, GBA and EKA respectively, from a shallow earthquake in
the Gulf of Suez, with mb = 5.1 (Earthquake 2 of Table 1). The
corresponding correlograms, indicating the arrival of phased
energy, are also shown - Fig. 6(b), (e) and (f). The second arrival
at CBA (labelled t l ) is tentatively identified as pP. On t h i s
assumption, realistic bounds on the amplitudes, and polarities
where unambiguous, are included in Fig. 6 for each station. Because
pP is not clearly seen at either YKA or EKA, an upper limit on its
amplitude is deduced from the noise amplitude alone. Any loss of pP
energy above the source is again assumed to be negligible. It is
important to realize that even with these very wide constraints on
possible relative amplitudes, each station places a substantial
constraint on the fault plane orientation. Table 3 shows the frac-
tion of orientation space acceptable, assuming the measurements
made in Fig. 6. The orientations compatible with a l l three
stations are plotted in Fig. 7. One of these orienta- tions (shown
with an arrowhead in Fig. 7) is used as input to the modelling
program, in order to test the reproducibility of other features of
the seismograms, such as the interfering waveform of the pP and sP
pulses. The model parameters used are shown in Table 4 and the
resulting theoretical seismograms are shown in Fig. 6(c), (f) and
(i). The match between theoretical and observed seismograms is
good, and it is seen that the third arrival at GBA (marked as t2 in
Fig. 6) models well as sP. Again, not all the orientations shown in
Fig. 7 exhibit this additional match of sP; in fact the absence of
any significant energy following P in the YKA seismogram indicates
that the null vector - Fig. l(c) - is oriented near to the upper
hemisphere take-off angle to YKA. However, even without sP
information, the assumption of a double-couple has allowed the
fault plane to be constrained to within 0.3 per cent of orientation
space, using three low signal-to-noise ratio seismograms for an mb
5.1 earthquake.
slip angle in fault plane t,) 5" 30" 60" 90" 120" 150" 180" + +
+ C h < , \ + + + + + + + + + + + + + + + + + + + ) + + + + + +
+ + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . .
- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
A < , + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + i
-
Fault plane solutions Table 4. Model used for Gulf of Suez
earthquake.
(a) Source parameters
Focal depth = 7.8 km Radius of fault plane = 1.0 km (circular)
Stress drop across fault = 100 bar Fracture velocity = 0.6 us Dip
of fault plane 6 = 15' Direction of slip in the fault plane I) = 2"
Strike of fault plane u = 96"
@) Source structure
P-wave velocity S-wave velocity km/sl km/s
Layer 1 6.0 3.1 Layer 2 6.1 - Layer 3 8.1 -
(c) Receiver structures
YKA and GBA as in Table 2; EKA - Parks (1967).
Density g cm-)
2.8 2.9 3.4
393
Thickness km
18.1 18.0 00
5 Comparison of the relative amplitude and first motion
methods
Since the relative amplitude method provides a means of directly
computing fault plane solutions, it is important to compare it with
the first motion method. Although no detailed comparison between
these methods is given here, several fundamental advantages of the
relative amplitude method are worthy of note.
The use of amplitude information enables the maximum source
information to be extracted from seismograms, even if the P-wave
first motion polarity cannot be read unambi- guously. The above
examples show that if a double-couple mechanism is valid (an
assump- tion widespread in first motion studies) then fault plane
solutions can be computed at much lower mb (- 5.0) and with far
fewer seismograms (as few as three) than is possible using the
first motion method. It follows that if many observations are
available, the large redundancy of information could be used to
confidently confirm or reject any assumed radiation pattern.
Moreover, because confidence limits on the amplitudes are
specified, it follows that a realistic assessment of the
reliability of each observation is implicit in the input data -
this is impossible to achieve with first motion observations. The
need to design Q posteriori weighting factors, based on the mutual
consistency of the data (e.g. Ingram 1959; Knopoff 1961) is thereby
entirely avoided. Barley & Pearce (in preparation) have made a
detailed comparison of fault plane solutions computed using both
methods for one earthquake.
Conclusions
Examples have demonstrated that the relative amplitudes and
polarities of P and pP can be used to provide a clear measure of
the uniqueness of fault plane orientations deduced by the modelling
of shallow earthquakes. As a means of computing fault plane
solutions directly, it has been shown that the relative amplitude
method is capable of providing a well- constrained fault plane
orientation using only a small number of P/pP observations, and
that in some cases SP can provide a further useful constraint on
the orientation. Several advantages of the relative amplitude
method over the first motion method have been noted; these suggest
that the relative amplitude method has a wider field of application
than that of first motions.
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394 R. G. Pearce In a future paper, the relative amplitude
method will be applied to 'a larger number of
earthquakes, and sP observations will be included in the
computations. By using long-period seismograms, which are affected
less by the effects of the propagation paths of each phase, the
extension of this method to the study of fault plane orientations
of deep focus earth- quakes will be considered.
Acknowledgments
I should like to thank Dr H. I. S. Thirlaway and his colleagues
for facilities provided at MOD Blacknest, where this work was
carried out. In particular, I thank A. Douglas and J. B. Young for
useful discussions. The Natural Environment Research Council is
acknowledged for financial support.
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