Geophys. J. R . astr. Soc. (1981) 64,163-185
Earthquake seismograms that show Doppler effects
due to crack propagation
A. Douglas
Ministry of Defence, Procurement Executive, Blacknest,
Brimpton, Reading, Berkshire RG7 4RS
J. A. Hudson
Department of Applied Mathematics and Theoretical Physics,
Silver Street, Cambridge CB3 9EW
P. D. Marshall Ministry of Defence, Procurement Executive, Blacknest,
Brimpton, Reading, Berkshire RG7 4RS
Received 1980 May 22
Summary. Examples are presented of earthquake P-wave pulses seen on
broadband seismograms, t o show that on such recordings the pulse shapes are
more clearly seen than on conventional short-period and long-period seismograms. Most of the broadband seismograms have been chosen because they
show marked differences between the pulse lengths of P and those of the
surface reflections. In addition some of the pulses appear to have smooth
onsets and abrupt trailing edges so that the onset of the pulse is difficult t o
observe and the largest amplitude arrivals seen on the seismogram coincide
not with the onset of motion but with the termination of motion: that is the
large arrivals mark stopping phases of motion.
We assume that the differences in pulse length are due to the effects of a
moving source - that is a Doppler effect - and that the pulses with smooth
onsets and abrupt trailing edges can be modelled simply by a source
propagating on a line with low radiation amplitude at the start of motion. A
trial and error method guided by a published fault plane solution is then used
to obtain a fit between observed and computed seismograms for one of the
earthquakes. This process leads to an estimate of the crack speed of about
1.4 times the S-wave speed.
The errors that may arise in estimating source depths and orientation, if
stopping phases are not recognized as such, is discussed.
1 Introduction
It is usual to assume that earthquakes are due to faulting: a crack initiates at a point and
spreads out to form a fault plane; as the crack passes a given point, slip takes place on the
fault plane resulting in a stress drop and the radiation of seismic waves. On such a model the
164
A . Douglas, J. A. Hudson and P. D. Marshall
seismic waves are radiated by a moving source and so the waves should show what Bollinger
(1 968) has called Doppler-type effects by analogy with similar phenomena in electromagnetic and acoustic wave propagation. If these effects can be observed then they can be
used to estimate the speed of crack propagation and the fault dimensions but, as Bakun,
Stewart & Bufe (1978) have pointed out, observations of such effects are surprisingly few.
The Doppler effect would be seen in body waves mainly as a variation with azimuth and
distance in the duration of the pulses radiated by a source. For a simple isotropic source
running on a line, for example, the pulse with the shortest duration and largest amplitude
is radiated in the direction of the propagating crack (assuming the crack speed is less than
the wave speed) whereas that with the longest duration and smallest amplitude is radiated in
the opposite direction (Berckhemer & Jacob 1965). For sources with more complicated
geometry not only WIU the duration of the pulse vary with direction relative to the fault
plane but the pulse shape will also change quite considerably (Berckhemer & Jacob 1965;
Savage 1966). (Doppler effects will also be seen in the surface wave radiation from earthquakes as shown, for example, by Press, Ben-Menahem & Toksoz (1962); in this paper,
however, we are concerned only with body waves.)
Most attempts so far to infer the motion and geometry of a fault have used spectral
methods in which the observed spectra of body wave pulses are compared with the spectra
calculated from the idealized model. In particular, the variation of the body wave spectrum
with azimuth and distance has been compared with the computed directivity function
(Ben-Menahem, Smith & Teng 1965; Hirasawa & Stauder 1965; Khattri 1972) w h c h is
related to the Doppler frequency shift. In other studies the observed spectrum corrected for
transmission path and instrumental response, has been compared directly with the synthetic
spectrum (see Davies & Smith 1968, for example). Probably the best known of these
methods is that w h c h utilizes the relationship between the corner frequency of the radiated
spectrum and the radiated pulse length, and hence source size and crack speed (Brune 1970,
1971).
Recently, Bakun et al. (1978) and Lilwall(l980) have found evidence of Doppler effects
in short period (SP) seismograms from local earthquakes. At teleseismic distances, however,
most observations of Doppler effects seem to have been made on long period (LP) P seismograms (Bollinger 1968, 1970; Berckhemer & Jacob 1968; Davies & Smith 1968; Khattri
1972). The use of LP rather than SP seismograms is necessary because with SP observations
it is usually difficult to separate the effects on pulse duration of anelastic attenuation from
those that are due to the moving source. However, if P,pP and sP observed at a given station
from a shallow earthquake are compared, then any differences in the duration and shape of
these pulses is most likely to be due to the effects of the moving source because attenuation
effects should be nearly constant for all three pulses; by comparing P and pP in this way
Douglas et al. (1974) and Langston (1978) have been able to show that possible Doppler
effects can be seen on some SP seismograms. For the seismograms studied by Douglas et al.
(1974) the Doppler effects can most easily be seen when the SP seismogram is passed
through an inverse filter which compensates for the effects of the seismograph to produce a
seismogram that displays ground displacement with constant magnification over a wider
band than is displayed in the original seismogram. This may explain why Doppler effects are
not often clearly observed on seismograms: most seismographs record only a narrow band of
frequencies and these pass bands may be too narrow to allow differences in pulse shapes to
be easily observed.
In this paper we present examples of seismograms to show how the shape and duration
of body wave pulses are more easily seen on broadband (BB) seismograms (which display
ground displacement with constant magnification over a band from about 0.1 t o 5 Hz) than
Earthquake seismograms due to crack propagation
165
on conventional SP and LP seismograms. Most of the BB seismograms shown here exhibit
differences in pulse length between P and the surface reflections ( p P or sP or both); a
particularly clear example of this is shown in Fig. 6(c). We hterpret such differences in pulse
length as the effects of a moving source. For one of the earthquakes studied we attempt to
estimate the speed of crack propagation.
2 Some theoretical models of earthquakes
In this section we illustrate the influence of the speed of crack propagation on the body
wave pulses radiated by some simple earthquake source models. We ignore any radiation
pattern effects that arise from the type of force system acting at the source.
Consider first a source of seismic waves running unilaterally on a line of length I at speed
u (< P) in a medium with P- and S-wave speeds of (Y and P, respectively. Assume that the
source activates instantaneously at each point and that the radiation into P and S waves is
A,f(t) and A,f(t) respectively at each point in time and is the same in all directions; the
source moves along the line x = ut (0 Q x Q I). The radiation in the far-field at distance r
from the origin is, for P:
)6(x
r
-
ut')f(t')dxdt',
(Y
where
u cos e
q a = 1 --,
(Y
r=t
r/a, and 0 is the angle between the line of the source and the direction of the
observer. Similarly the radiation for S is
-
where
u cos e
4 p = 1 -, r' = t
P
-
r/p.
Thus the amplitude and shape of each pulse varies with the crack speed. However, the area
under the curve remains
s_
m
(Ap/r)
f (0dt
and
m
fA,/r)
J- _ f ( t ) d t
for P and S respectively, whatever the value of u.
If f ( t ) is harmonic:
f ( t )= exp (- iwt)
A. Douglas, J. A. Hudson and P. D. Marshall
166
then
a,(?) = A , exp (- iw7/4&)
~
4LYr
which displays the usual Doppler shift of frequency. Alternatively, we see that the Fourier
spectrum of apis:
where ?is the Fourier transform off.
Consider now a stationary source of duration t o = l / u which radiates P and S pulses
A p f ( t ) and A , f ( t ) . Compared to this stationary source a moving source travelling towards
the observer (0 Q 8 < n / 2 ) radiates pulses with larger amplitude and shorter duration: in
the frequency domain the reduction in pulse length results in a shift in the corner frequency
and a general broadening of the spectrum t o higher frequencies. A moving source receding
from the observer (n/2 < 0 Q n ) shows the opposite effects.
So far we have assumed that the speed of the source is less than P. When u = the picture
remains the same except that when t9 = 0, 4a is zero and as(7‘) reduces to an impulse:
If p < u < a,up and a, are given by the same formulae except for 8 in the range cos 8 > Piu
where q p becomes negative; within this cone of ray directions the source outruns the shear
wave fronts generated at each part of the line and the observed pulse is the (scaled) mirror
image of the original :
We assume that the extension of the crack is self causative and so u G a ; the P pulse shape
will not be reversed at any value of 8.
Some of the effects of the speed of the source on the shape of the pulses are illustrated
in Fig. 1 w h c h shows examples radiated in the direction 8 = 0 for source speeds ranging
v=o 5 0
v=p
v.20p b + p r
V’O
Figure 1. The P and S pulses are functions of retarded time T radiated by a unilateral line source of length
I in the direction of source propagation for various crack speeds, u . It is assumed that the S-wave speed
13 = a/\/3 where a is the P-wave speed.
Earthquake seismograms due to crack propagation
167
from u = 0.50 to u = a. The radiation from the source is assumed to be proportional to x / l
for 0 G x G I and zero otherwise i.e. f ( t ) = ut/Z, 0 G t Q l/u and the total pulse length is T~ =
b a / u for P and 7,= Zlqpl/u for S. Note that when the source speed is 2ab(a to)-’ the P and
S pulses are of equal length (for 0 = 0) but are mirror images of each other.
For a bilateral line source in which the crack runs from an initial point at the centre of
the line in both directions to the ends, an observer would see the superposition of the pulses
from two unilateral sources. For example, an observer in line with the crack would see a P
pulse which has a short high-amplitude beginning from the approaching section of the crack
and a long low-amplitude tail from the receding section. An observer at a point in the plane
perpendicular to the crack (0 = 7r/2), however, would see the original pulse shape f ( t ) .
If an earthquake source can be approximated by a unilateral line source and if the
duration of any two pulses radiated by the source can be measured then (as pointed out by
Berckhemer & Jacob 1965) an estimate of u can be obtained if also 0 for each of the two
pulses, and a and 0,are known. For example, assume that P leaves the source in a direction
8 = 0 and pP in a direction 8 = 7r. Then R the ratio of the duration of P to the duration of
pP is given by:
R
= (a - u)/(.
+ u),
(la)
and so
u = a(1
-
R)/(l +R),
(1b)
assuming that the effect of damping on both pulses is the same. Given u and a measurement
of pulse duration 7p then Z can be found. From equation ( l b ) it can be seen that if R z 0.27
then u 5 0 assuming p = 4 4 3 . If the ratio of the duration of P to pP, therefore, (or of pP
to P) is less than 0.27 the crack speed must be greater than 0.
The amplitude of the radiation is of course modified in the case of a model earthquake
source by the radiation pattern of an idealized crack.
Consider now the two-dimensional source model of Savage (1966): in this model a crack
is assumed to be initiated at a point and to spread with uniform speed in all directions finally
coming to rest along a circular or elliptic contour. The crack may be initiated at either the
centre of the circle or ellipse or at a focus of the ellipse. Slip may occur instantaneously as
the crack passes or can follow some specified function of time. For the amplitude of the
displacement on the fault plane Savage (1966) considers two possibilities: in one the
amplitude is assumed to be constant over the fault: in the other the amplitude falls off
towards the edge of the fault as (1 - x ~ / s ~ )where
~ ’ ~ ,x is the distance of a point from the
centre of the ellipse and s is the distance of the edge of the fault from the crack centre along
a radius through the point.
The characteristics of the pulse shapes from Savage’s (1966) model of an elliptic or
circular fault are similar to those from a line source but rather more complicated. Let us
consider a circular fault with a crack moving out in a circle from the centre with speed
u(u < 0) and with uniform displacement over the whole fault plane. In Fig. 2 AOB is a crosssection through the fault plane. The crack starts at 0 and spreads out towards A and B. An
I
p
2
________A
I
0
6
Figure 2. Cross-section of fault plane with observers at right angles and in-line with the fault plane.
A. Douglas, J. A. Hudson and P. D. Marshall
168
P
,
0
_
_
_
~
-
pP
SP
~
L
____
I
I
Figure 3. Computed P seismograms for a circular fault in a half space. The centre of the fault plane is at
a depth of 25 km: IJ = 0.60:(a) radiated displacement; (b) convolution of displacement shown in (a) with
the impulse response of a short-period seismograph; (c) convolution of displacement shown in (b) with
the impulse response of a broadband seismograph; (d) cross-section through the source in the plane of the
observer.
observer at Pz sees the crack approaching along OA, receding along OB and moving with
intermediate speeds in other directions in the fault plane. The P pulse has therefore a steep
front, a high-amplitude beginning and a long low-amplitude tail. The pulse length is s(l/u +
l/a), where s is the radius of the circular fault plane. An observer at PI on the other hand
receives at any instant, radiation from a crack edge that grows at a constant rate proportional t o u. The leading edge of the pulse is therefore a ramp which drops to zero after a
time s/u. The pulse is shorter than at Pz, has a less steep front, and high amplitudes at the
rear rather than at the beginning. For observers at other positions the pulse shapes are
intermediate between the two extremes with lengths between s/u and s(l/u + I/&).
Elliptic faults generate pulses of a similar kind. If the minor axis is much shorter than the
major axis the radiation will clearly be similar to that from a bilateral line fault if the crack
initiates at the centre of the ellipse, or a unilateral line fault if the crack starts at a focus.
Figs 3 and 4 show examples of the pulse shapes radiated to long range from faults of the
type proposed by Savage (1966): the examples are taken from Douglas, Young & Hudson
(1973) and are shown to illustrate some of the effects of a moving source on the radiated
pulse shapes. The model used is a source at a depth of 25 km in a halfspace with P-wave
Earthquake seismograms due to crack propagation
169
I.?
I
Figure 4. Computed P seismograms from an elliptic fault in a half space with major axis of the ellipse
vertical and rupture initiation at upper focus. The upper focus of the fault plane is at a depth of 25 km:
u = O . 6 p : (a) radiated displacement; (b) convolution of displacement shown in (a) with the impulse
response of a short-period seismograph; (c) convolution of displacement shown in (b) with the impulse
response of a broadband seismograph; (d) cross-section through the source in the plane of the observer.
speed 6.7 km s-', S-wave speed 3.96 km s-' and density 2.9 g ~ m - The
~ . speed of crack
propagation is assumed to be 0.60, the take-off angle of P to be about 23" to the downward
vertical at the source (the take-off angle to travel to 60"). The displacement discontinuity
on the fault plane falls off to zero in the manner described above. The radiated signal
consists of P, pP and sP.
Fig. 3(a) shows the pulses radiated on an azimuth at right angles to the strike from a
circular dip-slip fault with fault plane dipping at 45'; the radius of the fault plane is 5 km.
The relationship of the ray paths on which P, pP and sP leave the source to the fault plane is
shown in Fig. 3(d). Direct P for the model is radiated in a direction that makes a relatively
small angle with the fault plane compared to pP so that the leading edge of the pulse is
steeper than the trailing edge. For pP the direction in which the pulse leaves the source is
more nearly at right angles to the fault plane than for P; the leading edge of pP is thus less
steep than the trailing edge. For sP, because the S-wave speed is much closer than the P-wave
speed to the crack speed the effects of the moving source in steepening the leading edge
of the pulse is again clearly seen even though the angle between the fault plane and the sP
ray path is quite large.
170
A. Douglas, J. A. Hudson and ?! D. Marshall
Fig. 3(b) and (c) show how the ground motion shown in Fig. 3(a) would be recorded on a
SP and a BB seismograph respectively; the magnification of these seismographs as a function
of frequency is shown in Fig. 6. Note that the BB seismograph, as one would expect, tends
to follow the ground displacement better than the SP. The SP seismograph shows two or
three arrivals associated with each pulse; these arrivals on the SP seismogram coincide with
those parts of the pulse where the rate of change of displacement is greatest. In general there
will be three such arrivals: one coincides with the start of the pulse, one with the end of the
pulse and the third marks the time when from the position of the observer the crack first
appears t o reach the edge of the fault (Savage 1966). Following Savage (1966) we can think
of the first arrival on the SP seismogram as marking the starting phase of motion at the
source and the second and third arrivals as marking stopping phases.
Fig. 4(a) shows the pulse radiated from an elliptic dip-slip fault dipping at 90"; the major
axis of the ellipse is vertical. The semi-major and semi-minor axes are 2.5 and 2.0 km
respectively, and the crack is assumed to initiate at the upper focus. The azimuth on which
the pulses are assumed to be radiated is again at right angles to the strike. The ray paths at
source are shown in Fig. 4(d). For direct P the source is effectively moving towards the
observer during the whole time the crack propagates. For pP and sP the crack spreads
towards the observer for only a short time (the time to travel from the upper focus to the
upper limit of the ellipse) which gives a steep front to the pulse. For the remainder of the
time, the crack is moving away from the observer so pP and sP are much broader than the
direct P pulse. Fig. 4(b) and (c) show the BB and SP seismograms that would be recorded for
such ground motion. Note again that the arrivals on the SP seismogram mark starting and
stopping phases of motion at the source.
Computed SP seismograms for the fault models of the type illustrated in Figs 3 and 4
always show prominent arrivals that mark the starting phase of the pulse because the
derivative of displacement with respect to time is discontinuous at the start of the pulse.
Consider, however, the radiation from a source running on a line where the slip is proportional t o ( x / Z ) ~ - (x/Z)~ for 0 < x Q I; x is the distance of the moving source from the
point of initiation. Then f(r) has the form { (t/to)3- (t/to)6)for 0 < t Q to= Z/u and the
observed P pulse is proportional to
This form of f ( t ) was chosen because the gradient of the displacement is continuous at the
start of the pulse whereas at the end of the pulse the gradient is discontinuous and as shown
in Sections 3 and 4 below there is some evidence that earthquakes do radiate pulses of a
similar type. The radiated pulse for this type of source for toqa = 1 s is shown in Fig. 5(a).
The bulk of the energy lies in the pass band of the BB seismograph where the magnification
is constant with frequency and the phase shifts are small so that the BB seismogram (Fig. 5b)
tends to follow ground motion. For the SP seismogram on the other hand most of the
energy in the pulse lies outside the pass band and the main arrival comes from the trailing
edge of the pulse where the short-period energy is concentrated. Consequently the SP
seismogram (Fig. 5c) shows an arrival marking the starting phase of motion that is small
relative t o the main arrival which marks the stopping phase. For a pulse of much longer
duration for which toqa = 10s (Fig. 5d) most of the energy in the pulse lies at low
frequencies, outside the pass band of the BB seismograph. In consequence, the most
prominent feature on the BB seismogram then marks the stopping phase of motion (Fig. 5e),
just as in the SP seismogram for the 1 s pulse. The examples of how the SP and BB seismo-
171
Earthquake seismograms due to crack propagation
n
Figure 5. (a) Ground displacement of the form f ( 7 / t o q , ) = ( ~ / r , q , ) ~- (7/f0q,J6for 0 Q T < f 0 q , as a
function of time T : r,q, = 1 s. (b) Convolution of displacement shown in (a) with the impulse response of
a broadband seismograph. (c) Convolution of displacement shown in (a) with the impulse response of a
short-period seismograph. (d) Ground displacement of the form shown in (a) but with t , q , = 10 s. (e)
Convolution of displacement shown in (d) with the impulse response of the broadband seismograph.
graphs respond to theoretical pulses with smooth onsets (Fig. 5) are used in Sections 3 and 4
to guide the interpretation of some observed seismograms.
3 Body wave pulse shapes radiated by earthquakes
Fig. 7 shows the SP and BB seismograms for five earthquakes (see Table 1 for details). The
BB seismogram of the Honshu earthquake was derived from the SP using the method
described by Douglas, Hudson & Barley (1981) and is the seismogram that would have been
recorded by a seismograph with the BB response shown in Fig. 6 except that an additional
Table 1. Earthquakes used.
Date
Region
Origin
time
1970 July 29
Burma-India
border region
Off coast of Chiapas,
Mexico
Near W. coast of
Honshu
Kodiak Island
Fox Island
North Sumatra
10:16:19.3
1971 August 20
1971 September 21
1973 March 22
1975 November 30
1975 December 17
Depth
(km)
Station
Distance mb
(degree)
59
Wolverton,England
74.1
6.5
21 :36 :09.6
33
Wolverton, England
80.4
5.8
08:43:21.9
186
5.4
18 :14 :37.2
20: 30: 17.0
05:35:17.8
38
24
17
Warramunga,
57.1
Australia
Wolverton, England 69.3
Wolverton, England 75.8
Wolverton,England 90.4
5.9
5.7
5.6
Hypocentre information and magnitudes are taken from the PDE cards published by the National Earthquake Information Service (NEIS).
0 01
I
I I
I
1 I l l 1
I
I
1
1
I l l l l
I
1
I 1 I I I l
I
I I
I
1 1 1 1 1 1
I
r
I
I
I 1 1 1 1
I
I
I
I I I l l
Frequency (Hz)
Figure 6. Relative magnification of seismographs. .- .- . -. Narrow band long-period seismograph,
__ world-wide standard station long-period seismograph, ----- broadband seismograph, . . . . . worldwide standard station short-period seismograph.
Wiener ftlter has been applied to attenuate low frequency noise. The BB seismograms of the
earthquake at Kodiak Island and off the coast of Mexico were recorded on a BB seismograph with response that differs slightly from that shown in Fig. 6 : the displayed seismograms
have been corrected for differences between the response of the seismograph used to record
the signals and the BB response shown in Fig. 6. The remaining BB seismograms were
obtained by integrating the recorded output of a seismograph that has a response that has a
constant amplification for ground velocity in the band between about 0.1 and 5 Hz; the
effective response for the integrated output does not differ significantly from the BB
response of Fig. 6.
In the seismograms from the North Sumatra earthquake (Fig. 7a, b), both direct P and
what are probably the surface reflections are visible in the record and it is clear that the
surface reflections are broader pulses than P and consequently contain relatively less highfrequency energy than P. The difference in frequency content of P and the later arrivals is
illustrated by comparing the relative amplitudes as seen on the two types of seismogram; on
the SP seismogram P is larger relative to the later arrivals than on the BB seismogram. The
difference in amplitude, pulse width and frequency content between P and the surface
reflections seem to be consistent with the model of a unilateral crack that propagates in a
direction that lies close to the direction of the P ray path at the source; the small amplitude of P relative to pP and sP is to be expected if the usual assumption is made that the
plane of the crack is a nodal plane of the double-couple radiation pattern for P.
The seismograms of the North Sumatra earthquake also show a small hgh frequency
pulse A, arriving just before the surface reflections. By comparison with the seismograms
from the Burma-India border earthquake discussed in the next section we suggest that this
pulse A may be an S to P conversion at the Moho beneath the source.
Turning now to the seismograms from the earthquake off the coast of Mexico (Fig. 7c, d):
the BB seismograms show P and what we interpret as either pP or sP or both. As with the
Earthquake seismograms due to crack propagation
173
Figure 7. Short-period and broadband seismograms of five earthquakes. (a) Short-period P-wave seismogram for earthquake of 1975 December 17 with epicentre in North Sumatra as recorded at Wolverton,
UK. (b) Broadband P-wave seismogram for earthquake of 1975 December 17 with epicentre in North
Sumatra as recorded at Wolverton, UK. (c) Short-period P seismogram for earthquake of 1971 August 20
with epicentre off west coast of Mexico as recorded at Wolverton, UK. (d) Broadband P seismogram for
earthquake of 1971 August 20 with epicentre off west coast of Mexico as recorded at Wolverton, UK.
(e) Short-period P-wave seismogram for earthquake of 1975 November 30 with epicentre in the Fox
Islands as recorded at Wolverton, UK. (f) Broadband P-wave seismogram for earthquake of 1975
November 30 with epicentre in the Fox Islands, as recorded at Wolverton, UK. (g) Short-period P-wave
seismogram for earthquake of 1971 September 21 with epicentre near west coast of Honshu as recorded
at Warramunga, Australia. (h) Broadband seismogram derived from (g). (i) Short-period P-wave seismogram for earthquake of 1973 March 22 with epicentre near Kodiak Island as recorded at Wolverton, UK.
(j) Broadband P seismogram for earthquake of 1973 March 22 with epicentre near Kodiak Island as
recorded at Wolverton, UK.
174
A. Douglas, J. A. Hudson and P. D. Marshall
North Sumatra earthquake the surface reflections are clearly of much lower frequency than
P and again it would seem that the seismograms indicate the general nature of the source was
a unilateral crack that propagated to increasing depth with time. Note that on the SP seismogram there is little evidence of the surface reflections.
The seismograms of the Fox Islands earthquake (Fig. 7e, f ) are not as easy to interpret
as those of the two earthquakes discussed above, however, it is clear that again the first
arrivals in the seismogram have much more high frequency energy than the later arrivals
which would suggest the source was also a crack that propagated to increasing depth with
time.
The seismograms for the Honshu and Kodiak Island earthquakes (Fig. 7g-j) are included
more because of the form of the pulses shown, than as a demonstration of possible Doppler
effects although the Honshu earthquake does appear to show such effects. The seismograms
from both earthquakes show among other arrivals P and pP Ignoring polarity differences
we see that the pulse shapes of pP (Fig. 7h) and P, pP (Fig. 7j) are very similar, in particular
the leading edge of these pulses appear to be less steep than the trailing edge so much so that
the onset of the P pulse on the BB seismogram for the Kodiak Island earthquake (Fig. 7j)
is difficult to pick. The first motion on the SP seismogram is consequently small relative to
second motion as is predicted by the results given in Section 2 where pulses with no discontinuity in gradient at the onset are convolved with the impulse response of the SP seismograph. The onset of pP on the seismograms from both the Honshu and the Kodiak Island
earthquakes may also have n o discontinuity in gradient but this cannot be observed because
of the presence of other arrivals. However, suppose we pick the onset of pP on the BB
seismograms at about To then it is clear that on the SP seismogram the largest amplitude of
pP marks the stopping phase of the pP pulse; in the absence of the BB seismogram most
analysts would probably pick the onset of pP at Tl, which is late by 1 s and leads to the
incorrect conclusion that the polarity is negative.
The seismograms of Fig. 7 demonstrate: (1) that wide differences between the duration
of P and the surface reflections are sometimes observed which, from visual inspection of the
seismograms, seem to be explicable as Doppler effects; and (2) that some earthquakes radiate
pulses sindar to those shown in Fig. 5 where the leading edge of the pulse has no discontinuity in gradient at the onset and is less steep than the trailing edge so that the largest
amplitude on the SP seismogram marks the end or stopping phase of the pulse. In the next
section we look in detail at some seismograms from a Burma-India border earthquake
which also appears t o show Doppler effects.
4 Burma-India border earthquake: 1970 July 29
Fig. 8 shows the P-wave seismograms from an earthquake on the Burma-India border (see
Table 1 for details) as recorded at Wolverton (WOL) in southern England on four seismographs with different responses: the relative magnifications of these seismographs as a
function of frequency are shown in Fig. 6. Two of the seismographs are the SP and BB
already described: the other two are a narrow band long-period instrument (NLP) with passband centred around 20s period and a long-period seismograph with the World Wide
Standard Station LP (WLP) response.
The broadband seismograms (Fig. 8c) show pulses Al, A2, A3 and A4, with A, and A4 of
longer duration than Al and A2. If A, is direct P and A3 and A4 are the surface reflections
pP and SPrespectively, these differences in the pulse duration appear to be evidence that the
elastic waves were radiated from a moving source. In this section we attempt to model these
effects t o see if the observed differences in pulse length can be explained in this way. We
Earthquake seismograms due to crack propagation
F\
175
\
-\
a
m
\
\
\
1
c!
I
lL1
%
2,'.
-_LI
Figure 8. P-wave seismogram recorded at Wolverton, England from the Burma-India border earthquake
of 1970 July 29. (a) Narrow band long-period seismogram, (b) world-wide standard station long period
seismogram, (c) broadband seismogram, (d) short-period seismogram.
have chosen to model this particular seismogram because the earthquake has been studied
by other workers and several fault plane solutions have been published (Molnar, Fitch & Wu
1973; Banghar 1974; Chandra 1975; Rastogi 1976). The nodal planes determined by Rastogi
(1976) using the P first motions and S wave polarization and those determined by Banghar
(1974) from P first motions are shown in Fig. 9(a). Rastogi (1976) has compared published
solutions and concludes that the differences between them are not significant. All the
solutions show a steeply dipping nodal plane striking roughly NNE-SSW. The other nodal
plane is not well determined but all the solutions agree that this nodal plane has a shallow
dip. As none of the solutions show a plunge of the null vector (the intersection of the two
nodal planes) of greater than 28" the first motion solutions can be summarized as indicating
the main component of faulting is dip-slip.
If we assume initially that the arrival Al is direct P, A3 is pP and A4 is SP then Al and A4
appear to have negative polarity and A3 positive. This, however, leads to a contradiction
when compared to published fault plane solutions because if one of the nodal planes is near
vertical this pattern of polarities can only arise from near strike-slip motion whereas the
fault plane solutions (Fig. 9a) predict mainly dip-slip motion.
Attempts to model the BB seismogram (Fig. 8c) show, however, that the arrivals A3 and
A4 probably mark stopping phases of pP and SP respectively and so in fact the polarity of pP
is negative and SP positive which is consistent with dip-slip motion on a steeply dipping fault
plane, as we now show.
'The evidence that A3 and A4 mark stopping phases comes from comparing the observed
seismograms (Fig. 8) with seismograms computed for a strike-slip fault (Model I: Fig. 9(b)
and Table 2 ) using the method of Hudson (1969a, b) and Douglas, Hudson & Blarney
(1972). The orientation of the fault plane was assumed to be the same as that of the steeply
dipping nodal plane of the fault plane solution of Banghar (1974) except that the dip was
taken to be 80" rather than 84". Assuming strike-slip the auxiliary plane was taken to be
vertical; the nodal planes are shown in Fig. 9(b). (Note that the nodal planes shown in
Fig. 9(b) are plotted for a source in a crust with P-wave speed 6.7 km s-l whereas the fault
plane solutions for the Burma-India border earthquake shown in Fig. 9(a) have been
176
A. Douglas, J. A. Hudson and P. D. Marshall
I
II
andIU
(b)
Figure 9. (a) Fault plane solutions obtained by Banghar (1974) and Rastogi (1976) for the Burma-India
border earthquake of 1970 July 29. (b) Nodal planes o f double couple sources used in an attempt to
compute theoretical seismograms to match the observed seismogram o f the Burma- India border earthquake.
obtained assuming that the source is in the upper mantle in material with P-wave speed of
about 8.1 kms-'.) An elliptical fault model of the type proposed by Savage (1966) was
used as the source: the major axis of the ellipse was assumed to be oriented in the direction
of maximum dip and the crack was assumed to be initiated at the upper focus.
As a guide to the choice of source dimensions and crack speed we start by assuming the
source is a simple line fault in the direction of maximum dip. For further simplification we
take the P pulse (arrival A,) to leave the source in a direction which is in line with the
approaching crack (0 =0) and the pP pulse (arrival A3) to leave in the opposite direction
barthquake seismograms due to crack propagation
177
Table 2. Some details of source models used for which theoretical seismograms have been
computed.
Source crust*
Crack speed (km s-')
Source model
Shape of fault plane
Dimensions of fault plane (km)
Depth (km) of focus
(point of initiation of rupture)
Model I
Model I1
Model 111
A
B
5.0 (fi 1.250)
Savage (1966)
Ellipse
6.75 X 3.375
(semi-axis)
72
C
5.5 (e1.380)
Line source
Line
39
(total length)
38
2.4 (e0.6p)
Savage (1966)
Ellipse
3.0 X 1.5
(semi-axis)
71
*Crustal models A, B and C are given in Table 3.
(0 = n). The observed value of R from Fig. 8(c) appears t o be about 0.5 so for 1y =
6.7 km s-l this gives a value of u (using equation 1b) of about 2.2 km s-'. This estimate of u,
however, is likely to be on the low side because of the simplifications made, so we use
2.4kms-'; that is 0.6& the relative crack speed used by Douglas et al. (1974). With this
value of u and taking the duration of A3 to be about 3.5 s then the estimated length of the
source is about 6 km. In our more detailed model, the major axis of the ellipse was thus set
t o 6 km and the minor axis arbitrarily to half this.
The crustal model used for the source region is based on the standard model of a continental crust with sediment (McEvilly 1964); the thicknesses of the layers were simply
adjusted to give the required delay-time between P and the surface reflections. For the
crustal structure at the receiver (Crust D, Table 3) we have taken the crustal model for
Eskdalemuir, Scotland (Parks 1967) and added a layer 1 km thick to simulate the sedimentary layer that is present in the vicinity of station WOL.
Fig. lO(a) shows the ground displacement computed for this model with strike-slip
motion on the fault; the sequence of polarities shown by P, pP and sP are the same as shown
by Al, A3 and A4 (Fig. 8c). However, the fit between the computed seismograms (Fig.
lob-e) and the observed seismograms (Fig. 8) is poor. Note, however, that if the section of
the computed WLP seismogram (Fig. 1Oc) between B, and B2 is inverted it is of similar shape
to the corresponding section of the observed WLP seismogram. This suggests that the true
polarities of pP and sP are the opposite of the apparent polarities seen on the observed BB
seismogram (Fig. 8c) and that the Burma-India border earthquake was indeed due to dipslip motion as shown by the fault plane studies.
Fig. 11 shows the computed seismogram for a dip-slip source (Model 11: Fig. 9b, Table 2).
There is now a better fit between the computed seismograms (Fig. 1lb-e) obtained for the
dip-slip source and the observed seismograms than obtained for the strike-slip source: in
particular the section between A3 and A4 of the observed WLP seismogram (Fig. 8b) is well
modelled.
The theoretical seismograms (Figs 10 and 11) suggest an explanation of arrival A2, an
arrival which so far we have ignored. The computed seismograms show an arrival at about
the time of A2 which is an S to P conversion at the Moho in the region of the source. The
converted wave has an amplitude similar to P for two reasons: (I) the P r a y path lies close to
a node of P whereas the S ray path is not as close to a node of S; and (2) the maximum S
amplitude for a double-couple source is over five times larger than the maximum P
amplitude; so although the conversion coefficient of S to P at the Moho is small the incident
S amplitude is large relative t o P.
The computed ground motion for the strike-slip model (Fig. 10a) shows an S to P
conversion that has shorter duration than direct P a s is expected when the crack speed is less
178
A. Douglas, J. A. Hudson and R D. Marshall
PP
Figure 10. Theoretical seismograms computed for a strike-slip source and a Savage (1966) fault model
(Model I : Fig. 9b and Table 2). (a) Ground displacement, (b) narrow band long-period seismogram, (c)
world-wide standard station long-period seismogram, (d) broadband seismogram, (e) short-period seismogram.
than 0.On the observed seismograms direct P (arrival A , ) and the possible S to P conversion
(arrival A,) seem to be about the same duration; if A, is in fact an S to P conversion then
this can only occur for P and S waves that leave the source on nearly the same ray path if
the crack speed is greater than the S-wave speed. Assuming a unilateral line fault on which
the crack is travelling towards the observer then the P and S pulses will have the same
duration (but will be reflections of each other in the T = O axis) if the crack speed is
2cu0/(cu t 0) which for the crustal structure assumed here is very close to 5.0 km s-'; this
crack speed was therefore used in computing the seismogram (Fig. 11) for the dip-slip
model (Model 11: Fig. 9b and Table 2). The dimensions of the fault plane have been chosen
t o give pulses of about the correct absolute duration.
The main discrepancy between the seismogram computed for the dip-slip source (Fig. 11)
and the observed seismograms (Fig. 8) is the strong negative motion that marks the onset of
pP (C, Fig. 1 Id): in other words the arrival that marks the start of p P i s too prominent. In
a partial attempt to reduce this starting phase we have replaced the ellipse with the simpler
unilateral line fault of the type discussed earlier where the slip is proportional to ( x / Z ) ~ ( x / Z ) ~ for 0 Q x G Z where x is the distance of the crack tip from the point of initiation; we
have already seen in Section 3 that pulses similar to those radiated by such a line source are
Earthquake seismograms due to crack propagation
J
d -
-
’q\
I”\
’
-
/
\
I
Illi
-~
i(1 ’
vl‘
I/
11
4---
-.“r
-
‘.
I
C’
e -
179
’1
h p ! ,,
~
;J-
1‘1
[\?/,
I
-si
/yp&+<\
LP”
Figure 11. Theoretical seismograms computed for a dip-slip source and a Savage (1966) fault model
(Model 11. Fig. 9b and Table 2). (a) Ground displacement, (%) narrow band long-period seismogram,
(c) world-wide standard station long-perlod seismogram, (d) broadband selsmogram, (e) short-period
seismogram.
seen on seismograms. Fig. 12 shows a comparison of the observed seismograms and seismograms computed using the unilateral line source; other details of the model used in this
computation are given in Table 2: Model 111.
The main discrepancy between the observed and computed seismograms (Fig. 12) is the
large amplitude arrivals more than 40s after onset which are shown by the observed NLP
and WLP seismograms but not the computed ones. However, it is clear that for the first 40 s
after onset and for the frequency range of about 0.05-1 Hz the computed seismograms
reproduce many of the major features of the observed. Moreover, the agreement is achieved
using a source orientation that does not differ significantly from published fault plane
solutions based on first motion polarities and S polarization. There is thus strong evidence in
support of our suggestion that the differences in duration and shape of the pulses shown on
the BB seismogram of the Burma-India border earthquake are due to the effects of a
moving source.
It is difficult t o give a measure of the uncertainty in the estimate of crack speed (and
fault length) obtained and more detailed work is required to develop a better fault model,
however there seems to be little doubt that the crack speed is greater than 0. For if it is only
the stopping phase of pP that is seen on the BB seismogram then the actual duration of pP
180
A. Douglas, J. A. Hudson and P. D. Marshall
J
0
10
20s
w
Figure 12. Comparison of observed seismograms and theoretical seismograms computed for a dip-slip
line source (Model 111: Fig. 9b and Table 2). (a) and @) observed and theoretical narrow band longperiod seismograms respectively, (c) and (d) observed and theoretical world-wide standard station longperiod seismograms respectively, (e) and (f) observed and theoretical broadband seismograms respectively,
(g) and (h) observed and theoretical short-period seismograms respectively.
must be greater than the apparent duration observed. The above analysis implies that the
crack speed must be greater than about 0.60, but the closer the crack speed is to 0 the
narrower should be the arrival A2 if this is an S to P conversion. So it would appear that only
by having the crack speed greater than 0 is it possible to have both the required ratio for the
duration of P and pP and Al and A2 of about equal width.
5 Discussion and conclusions
If we are correct in attributing the differences in the length and shape of the pulses of P
and the surface reflections pP and sP to Doppler effects then for some of the seismograms
Earthquake seismograms due to crack propagation
181
Table 3. Crustal structures used for modelling.
P-wave speed
(km s - ' )
S-wave speed
(km s - I )
Density
Thickness
(g ~ m - ~ ) (km)
3.0
6.1
6.4
6.7
8.15
1.66
3.5
3.68
3.94
4.75
2.35
2.7
2.9
2.9
3.3
2.0
19.0
24.0
38.0
3.0
6.1
6.4
6.7
8.15
1.66
3.5
3.68
3.96
4.75
2.35
2.7
2.9
2.9
3.3
3.0
19.0
20.0
50.0
3 .O
6.1
6.4
6.7
8.15
1.66
3.50
3.68
3.94
4.75
2.35
2.7
2.9
2.9
3.3
2 .o
15.0
15.0
46.0
3 .O
6.14
7.28
8.09
1.66
2.35
2.8
3.2
3.38
1.o
5.3
19.7
Oust A
Layer 1
Layer 2
Layer 3
Layer 4
Halfspace
m
Oust B
Layer 1
Layer 2
Layer 3
Layer 4
Halfspace
Oust
m
c
Layer 1
Layer 2
Layer 3
Layer 4
Halfspace
m
Oust D
Layer 1
Layer 2
Layer 3
Halfspace
-
m
Where the S-wave speed p is not specified it is assumed t o be a/J3 where
01 is the P-wave speed.
presented in this paper these effects seem to be much stronger and much more obvious than
those seen on other published seismograms. This is in part due to the fact that the effects
are more clearly seen on the BB seismograms which are not commonly recorded, which
suggests that if BB seismograms were more widely used strong Doppler effects would be
more often recognized than at present. We have not made any systematic search for seismograms that show obvious Doppler effects but our impression is that about 1 in 10 of BB
seismograms show such effects.
In this paper a trial and error method guided by published fault plane solutions is used t o
obtain a fit between observed and computed seismograms and as a consequence obtain
estimates of crack speed and length. Given BB seismograms, a more systematic method of
obtaining such estimates would be to combine the method of determining source orientations proposed by Pearce (1977, 1979, 1980) with that for determining fault geometry and
crack speed proposed by Bollinger (1968). This combination of methods should define a
range of possible source models and these can be checked and refined by comparing the
seismograms computed for such models with the observed seismograms.
The method of Pearce (1977, 1979, 1980) finds all fault plane solutions compatible with
the polarities and relative sizes of P and surface reflections on the assumption that the only
effect of the source on the relative sizes of the phases is that due to the radiation pattern.
182
A. Douglas, J. A. Hudson and I? D.Marshall
If P and the surface reflections are pulses of similar duration then the effects of the radiation
pat tern can be estimated (with error bounds) for simple amplitude measurements taken from
the seismogram. For seismograms with strong Doppler effects the amplitude of an arrival
depends on both the effect of the source radiation pattern and the moving source. However,
it should be possible to isolate the main effect of the radiation pattern by using the areas
under the pulses on BB seismograms rather than straight amplitudes for as pointed out in
Section 2 these areas are not affected by the moving source. Care would of course be
necessary in such a study to ensure that stopping phases were not mistakenly identified as
complete pulses although if such erroneous observations were used this would in general be
revealed because the observations would be shown to be incompatible (assuming that the
source has a double-couple radiation pattern). Ideally of course the seismograms used would
be from a seismograph with pass band that extends t o much lower frequencies than used
here. Such an instrument would record more nearly true ground motion than the BB seismograph we have used and so the possibility of error would be reduced.
For some or all of the fault plane orientations found to be compatible with the observations a systematic search could then be made in a similar way to that proposed by Bollinger
(1968) t o find combinations of crack speed and length (for faults of simple geometry) that
are compatible with the durations of the observed pulses. Lilwall(l980) has in fact successfully used a systematic method like that outlined above to analyse seismograms of local
earthquakes; we now intend to carry out a similar analysis using BB seismograms recorded at
teleseismic distances.
Most estimates of crack speed obtained from earthquake studies seem to be less than 0:
some published estimates obtained from body wave analyses are catalogued in Table 4. Of
these estimates only those of Lilwall (1980) are greater than 0: for the remainder the
estimates range from around 0 down to speeds of 0.50 or less. (Note, however, that some
estimates of P and S corner frequencies appear to imply speeds of crack propagation greater
than 0,Burridge 1975.) For low crack speeds Doppler effects will be small. When strong
Doppler effects are seen this implies that the crack speed is around the S-wave speed or
greater as shown by our study of the Burma-India border earthquake where the estimated
crack speed is about 0 . 8 (;~
. ~1.380).
Crack speeds between the shear wave speed and compressional wave speed have been
shown to be possible in two-dimensional numerical simulations by Andrews (1 976). Under
plane strain conditions and an imposed shear stress at infinity the (mode 11) crack moves
with increasing speed and appears to jump from the Rayleigh wave speed t o a speed of
about 4 2 0 . This sequence of events has been confirmed analytically by Burridge, Conn &
Table 4. Some published estimates of crack
speed derived from studies of body waves
from earthquakes.
Crack speed
5’-wave speed
AuthoI
0.5 1-0.62
0.45
0.67-1.0
0.77
0.50-0.67
0.60
0.51-0.86
1.73
Bollinger (1968)
Berckhemer & Jacob (1968)
Davies & Smith (1968)
Bollinger (1970)
Khattri (1972)
Douglas el al. (1974)
Langston (1978)
Lilwall(1979)
-
Earthquake seismograms due to crack propagation
183
Freund (1979). Our estimated crack speed of 1.38(3is @erhaps fortuitously) very close to
the calculated speed 4/20 which appears to be a stable value for subsequent cracking.
Mikumo & Miyatake (1978) have used three-dimensional models of faults and find that
where the static friction is homogeneous or weakly non-uniform over the fault plane the
crack propagates elliptically having the P-wave speed along the direction of pre-stress and the
S-wave speed at right angles t o this; when the static friction is heavily non-uniform the crack
propagates in an irregular manner. The high crack speed estimated for the Burma-India
border earthquake is thus not ruled out on theoretical grounds and Burridge (1973) reports
that crack speeds of greater than 0 have been measured in the laboratory. Detailed study
may show that in fact models of the type studied by Mikumo& Miyatake (1978) best fit
the observations for the Burma-India border earthquake. Rastogi (1976) has attempted t o
measure the crack speed for the Burma-India border earthquake using surface waves and
concludes that there is evidence that the crack propagated horizontally on the fault plane
from north to south; the fault length (horizontal) is estimated to be about 35 km and the
rupture speed about 3 km s-'. If the earthquake was due to a crack propagating down dip at
around the P-wave speed and horizontally at the S-wave speed as suggested by the Mikumo&
Miyatake (1978) model, then estimates of crack speed obtained from surface waves would
measure a rupture speed of around the S-wave speed and estimates obtained from teleseismic
recordings of body waves would be around the P-wave speed. This suggests how it may be
possible to reconcile the apparent conflict between the estimates of the crack speed obtained
by Rastogi (1976) and that obtained here. An earthquake source that propagated down dip
at the P-wave speed and horizontally at the S wave speed might also help to account for the
low frequency arrivals seen on the WLP seismogram of the Burma-India border earthquake
(Fig. 12c) and which are not accounted for by the simple line source model.
The seismograms shown in the paper illustrate some of the problems of making accurate
measurements. Some of the pulses have onsets that rise smoothly from the noise background
and it is impossible to pick out exactly where the phase starts. If the pulse shape observed at
each station is constant, errors in measuring the onset times might be systematic but if the
pulse shapes vary from station to station because of Doppler effects then the accuracy with
which an onset is measured will also vary and this will introduce scatter into travel-time
observations perhaps accounting for some of the observed error in travel time observations.
Where the onset on the BB seismogram emerges smoothly from the noise the first motion
on the SP seismogram is likely t o be small relative to later motion. This may lead to the first
motion being missed and the onset time and polarity a t the onset being read incorrectly.
It is well known that consistent first motions are difficult to obtain from SP seismograms; if
the polarities of first motion are read incorrectly then the onset times must also be in error.
The difficulty of measuring onsets and polarities is most acute for pP and sP where the
onset may be small and obscured by other earlier arrivals and where the largest amplitudes in
the arrivals may mark stopping phases of motion at the source. Only by combining first
motion data from many stations and by the use of models was it possible to demonstrate
that the apparent pP and sP arrivals from the Burma-India border earthquake probably
mark stopping phases.
Note that the arrivals seen between 20 and 40 s after P on the computed SP seismogram
(Fig. 12h) obtained from Model Ill (Table 2) are not true pP and sP but are generated by the
reflection at the Moho of downward radiated P and S waves followed by reflection at the
free surface. If such arrivals are present on observed SP seismograms and are assumed to be
pP and sP this will lead to an erroneous estimate of depth of focus.
It is our experience from attempting to interpret P-wave seismograms using modelling
that apart from P, pP and sP and the other near surface reverberations the most easily
184
A. Doughs, J. A. Hudson and P. D. Marshall
identifiable and most important arrival seen in computed seismograms is often the S to P
conversion at the Moho. For sources in the crust it is likely that this arrival will be the first
observable arrival for ray paths that leave the source close to a node in the P radiation
pattern. The polarity of the S to P conversion does not change sign in going through a P
node and so this may explain some of the errors in first motion observations near P nodes
for it may not be P but the S to P conversion that is read. If the S to P conversion can be
clearly observed it may provide a better estimate of the S pulse radiated by the source than
direct S because the anelastic attenuation of direct S is usually so great that only the low
frequency energy in the S wave pulse can be observed at teleseismic distances. Comparing
direct S with the S to P conversion may provide a method of estimating the relative attenuation of S and P waves. The time difference between the onset of P and the onset of the S
to P conversion may also be useful in estimating the depth of the Moho below the source.
References
Andrews, D. J., 1976. Rupture velocity of plane-strain shear cracks, J. geophys. R e x , 81,5679-5687.
Bakun, W. H., Stewart, R. M. & Bufe, C. G., 1978. Directivity in the high frequency radiation of small
earthquakes, Bull. seism. SOC.Am., 68, 1253-1263.
Banghar, A. R., 1974. Mechanism of earthquakes in Albania, China, Mongolia, Afghanistan and BurmaIndia border, Earthquake Notes, XLV, 13-25.
Ben-Menahem, A., Smith, S. W. & Teng, T., 1965. A procedure for source studies from spectrums of
long-period seismic body waves, Bull. seism. Soc. Am., 55,203-235.
Berckhemer, H. & Jacob, K. H., 1965. Synthetic seismic pulses from propagating faults, Berichte des
Institutes fur Meteorologie und Geophysik der Universitate FrankfurtlMain, Number 7.
Berckhemer, H. & Jacob, K . H., 1968. Investigation of the dynamical process in earthquake foci by
analysing the pulse shape of body waves, Berichte des Institutes fur Meteorologie und Geophysics
der Universitat FrankfurtlMain, Number 13.
Bollinger, G. A,, 1968. Determination of earthquake fault parameters from long-period P waves, J .
geophys. Res., 73, 785-807.
Bollinger, G. A., 1970. Fault length and fracture velocity for the Kyushu, Japan earthquake of October
3,1963, J. geophys. Res., 75, 955-964.
Brune, J. N., 1970. Tectonic stress and the spectra of seismic shear waves from earthquakes, J . geophys.
Res., 75,4997-5009.
Brune, J . N., 1971. Correction, J . geophys. Res., 76, 5002.
Burridge, R., 1973. Admissible speeds for plane-strain self-similar shear cracks with friction but lacking
cohesion, Geophys. J. R. astr. Soc., 35,439-455.
Burridge, R., 1975. The effect of sonic rupture velocity on the ratio of S to P corner frequencies, Bull.
seism. SOC.Am., 65,667-675.
Burridge, R., Conn, G. & Freund, L. B., 1979. The stability of a rapid mode I1 shear crack with finite
cohesive traction, J . geophys. Res., 84, 2210-2222.
Chandra, U., 1975. Seismicity, earthquake mechanisms and tectonics of Burma, 20" N-28" N , Geophys.
J . R. astr. Soc., 40, 367-381.
Davies, J . B. & Smith, S. W., 1968. Source parameters of earthquakes and discrimination between earthquakes and nuclear explosions,Bull. seism. Soc. Am., 58, 1503-1517.
Douglas, A,, Young, J. B. & Hudson, J . A , , 1973. Theoretical P wave seismograms from earthquake
source models, A W R E Report No. 0 50/73, HMSO, London.
Douglas, A., Hudson, J. A., Marshall, P. D. & Young, J. B., 1974. Earthquakes that look like explosions,
Geophys. J. R. astr. Soc., 36, 227-233.
Douglas, A., Hudson, J. A. & Blarney, C., 1972. A quantitative evaluation of seismic signals at teleseismic
distances - 111. Computed P and Rayleigh wave seismograms, Geophys. J . R. asfr. Soc., 28, 385410.
Douglas, A., Hudson, J. A. & Barley, B. J., 1981. Complexity of short period P seismograms, Geophys.
J. R. astr. Soc., submitted.
Hirasawa, T. & Stauder, W., 1965. On the seismic body waves from a finite moving source, Bull. seism.
Soc. Am,, 55, 237-262.
Earthquake seismograms due to crack propagation
185
Hudson, J. A., 1969a. A quantitative evaluation of seismic signals at teleseismic distances - I. Radiation
from point sources, Geophys. J. R. astr. SOC.,18,233-249.
Hudson, J. A., 1969b. A quantitative evaluation of seismic signals at teleseismic distances - II. Body
waves and surface waves from an extended source, Geophys. J . R . astr. SOC.,18, 353-370.
Khattri, K., 1972. Body wave directivity functions for two-dimensional fault model and kinematic
parameters of a deep focus earthquake, J . geophys. Res., 77, 2062-2072.
Langston, C. A., 1978. The February 9 , 1971 San Fernando earthquake: A study of source finiteness in
teleseismic body waves, Bull. seism. SOC.Am., 68, 1-29.
Lilwall, R. C., 1980. Fault mechanisms and subcrustal seismic velocities o n the Mid-Atlantic Ridge,
Geophys. J. R. astr. Soc., 60, 245-262.
McEvilly, T. V., 1964. Central US crust-upper mantle structure from Love and Rayleigh wave phase
velocity inversion, Bull. seism. Soc. Am., 54, 1997-2015.
Mikumo, T. & Miyatake, T., 1978. Dynamical rupture process on a three-dimensional fault with nonuniform frictions and near field seismic waves, Geophys. J . R . astr. Soc., 54, 417-438.
Molnar, P., Fitch, T. J. & Wu, F., 1973. Fault plane solutions of shallow earthquakes and contemporary
tectonics in Asia, Earth planet. Sci. Lett., 19, 101-1 12.
Parks, R., 1967. Seismic refraction networks in the study of the Earth’s crust, PhD thesis, University of
Edinburgh.
Pearce, R. G., 1977. Fault plane solutions using relative amplitudes of P and pP, Geophys. J. R. astr.
Soc., 50, 381-394.
Pearce, R. G., 1979. Earthquake focal mechanisms from relative amplitudes of P , pP and sP: method and
computer program, AWRE Report No. 0 41/79, HMSO.
Pearce, R. G., 1980. Fault plane solutions using relative amplitudes of P and surface reflections: further
studies, Geophys. J . R. astr. Soc., 60,459-487.
Press, F., Ben-Menaham, A. & Toksoz, M. N., 1961. Experimental determination of earthquake fault
length and rupture velocity, J. geophys. Res., 66,3471-3485.
Rastogi, B. K., 1976. Source mechanism studies of earthquakes and contemporary tectonics in Himalayan
and Nearby regions, Bull. Int. Inst. Seism. Earthqu. Engng, 14, 99-134.
Savage, J . C., 1966. Radiation from a realistic model of faulting, Bull. seism. SOC.Am., 56, 577-592.