Geophys. J. R.usfr. SOC.(1971) 23,451-460.
The Relative Excitation of Seismic Surface and Body
Waves by Point Sources
A. Douglas, J. A. Hudson and V. K. Kembhavi
(Received 1971 March 15)
Summary
Experimental evidence shows that earthquakes can be differentiated from
explosions OD teleseismic records by their relatively greater amplitude of
Rayleigh waves compared with the amplitude of the direct P-wave; in
effect the ratio of magnitude estimates mblMs, is greater for explosions
than for earthquakes.
In this paper we have tried to assess the contribution of source
mechanism alone to this effect. We have used point dilatation and
double-couple sources to represent explosions and earthquakes and
derived the amplitudes of the waves generated from such sources in a
uniform half space. The results show that source mechanism is an
important factor governing the value of mb/Ms.
1. Introduction
One of the principal methods of discriminating between earthquakes and explosions
lies in the measurement of the relative magnitudes of body and surface waves
(U.K.A.E.A. 1965; Marshall 1970). This is based on empirical evidence that the
magnitudes calculated from surface wave amplitudes of explosions are significantly
less than the calculated M , of earthquakes which have the same magnitude mb,
calculated from body waves.
The method has been shown empirically to work well for events with mb > 4
approximately, and on physical grounds one would expect the greater source size
of an earthquake of large magnitude relative to that of an explosion to give rise to
exactly this effect. Interference between seismic signals radiated from different points
on a fault would reduce the shorter periods more than the longer. Thus body wave
amplitudes would be reduced more than the longer period surface wave amplitudes.
Berckhemer (1962) supports this conclusion, giving evidence that larger earthquakes
give rise to longer periods in the spectrum of radiation.
However, since the source mechanisms of the two types of event also differ, it is
interesting to study the effect of this first before going on to look at the result of
increasing the size of the source. Therefore, we have calculated the relative amplitudes
of both types of wave radiated from double couple and dilatation point sources
respectively, the sources being situated in a uniform half-space.
2. Amplitude of body waves
We shall calculate the spectral amplitudes of body waves, rather than the amplitude
of the pulse in time under the assumption that, if the body waves are recorded on a
narrow-band seismograph, the amplitude of the trace is approximately the same as
45 1
452
A. Douglas, J. A. Hudson and V. K. Kembhavi
the spectral amplitude at the peak frequency. If we then compare amplitudes of
body waves from two different sources by calculating their ratio, the effect of the
recording instrument cancels out.
I. N. Gupta (1966) showed how one may use the reciprocity theorem to calculate
the amplitudes in the far-field radiation from point sources in a half-space from
expressions for the reflection of plane waves at a free surface. The argument, which
is extended here to apply to double couple and dilatation sources, goes as follows.
Let ul(x, t), u2(x, t ) be the displacement fields due to body forces f '(x, t ) and
f ,(x, t ) respectively, acting in a half-space. The elastodynamic reciprocal theorem
(Graffi 1947) states that
1
fi'(x,o).f2(x,o)dV
=
1
ii'(x, o).r2(x, o ) d V
(2.1)
V
V
where the bar denotes the Fourier transform in time and the integration is over the
whole half-space.
We now set up spherical polar co-ordinates R, 8,4 with origin on the surface of
the half-space and the axis 8 = 0 vertically downwards into the medium. If we take
f2 to be a point force acting in the R-direction at a point (R,, 8, 4,), a large distance
from the origin, the left-hand side of equation (2.1) becomes
E1R(R2: 8,942,
4,
(2 * 2)
the &component of ul.
If the point force is sufficientlyfar away the displacementsu2in the neighbourhood
of the origin will consist of plane waves. Moreover, since the direction of the force
is a node for the S-wave radiation pattern, the displacements will be due simply to
an incident plane P-wave, from the direction (8,,4,) and its reflections at the free
surface. Hence, in terms of Cartesian co-ordinates (x1,x2,x3) with the positive
x,-axis along 0 = 0 and the xl-axis lying in the plane 4 = 0,
[4 sin 0,
(fix3 cos e, 1
io
iw
io
+Al exp (- y x3 case, 1) - - coso,' A, exp (- - x ,
B
B
i i , ~= A cos 4, exp
(fi
x, sin 8),
a
(exp
a
C O S ~ , ~ ]
ii,, = ii,, tan+,
: [ cos e, (exp (2 cos e, 1
case,)) - sine; A,exp (- - x 3
B
B
ii32 = A exp ( E x , sin 8),
a
- A , exp
(- -x3
io
a
I
i
(2.3)
a x3
io
io
coso;)]
J
where x, = x1 C O S + ~ + X , sin4,, A is an amplitude factor, and A,, A, are reflection
coefficients;
A, =
A2
=
sin 28, sin 28,'
sin 28, sin 28,'
- (a2/B2)cos228,'
+ (a2/p2)cos228,'
-2 sin 28, cos 28,'
sin 28, sin 28,' (a2/B2)cos228,'
+
*
453
Relative excitation of seismic surface and body waves
Oz' is found by Snell's law
sine2
---
- sin&'
B
a
.
We now take the body force f' to correspond to a double-couple point source of
magnitude Q ( t ) acting at depth h below the origin,
I
j = 1,2,3,
where n, f are unit vectors normal to the null planes.
On substituting this into equation (2.1) we get
where E;,(x, t ) is the shear strain related to co-ordinates parallel to f and n due to
displacements uZ(x,t).
If, on the other hand, we take f' to correspond to a dilatation source of magnitude
Q'(t) acting at depth h',
j = 1,2, 3.
and so
G'(RZ, 02, 4 2 , 4 = el(o>[div~z(X,
Wl,=(O,
0, W).
The ratio of the two-body wave amplitudes is (dropping the subscript 2)
a'
[ re+
+Az exp - i o h 2P
a
x
I
ioh'
I+A' exp (-2-cose)
a
COS~')]
B
I
-1
,
(2 * 8)
454
A. Douglas, J. A. Hudson and V. K. Kembhavi
where
The terms with the factor exp (-2iwh cos O/a) clearly represent the free surface
reflections p P of dilatation waves from the source. The term with the factor
exp [ -ioh(cos 8/a cos O’/B)] represents the reflection sP of a shear wave from the
double-couple source (there is no corresponding wave from the purely dilatational
source).
+
3. The relative amplitudes of surface and body waves
The ratio of the spectral amplitudes of Rayleigh waves from double couple and
dilatational sources may be found in the same way as x p above, using a reciprocal
theorem for surface wave displacements alone (Douglas, Hudson & Kembhavi 1971).
If the source magnitudes are again Q(t) and Q’(t) and the source depths h and h’,
respectively, we have for this ratio,
where C, is the Rayleigh wave speed and
va
= ( 1 - c ~2/a2) t
qfl
(1-cR2/Bz)*.
Again, the effect of the recording instrument cancels out in the ratio xR. There
are no dispersion factors to take account of as the half-space is homogeneous.
If tbe source is sufficiently deep, the surface reflections p P and sP do not arrive
until the main part of the direct body wave signal has passed. In this case, the
magnitude mb of the event is calculated from the amplitude of the recording of the
direct P-wave only. The ratio of the measured maximum peak-to-peak amplitudes
of body waves from double couple and dilatation sources is therefore
where cop is the peak frequency of the recording instrument, 8 is the angle of emergence
of the ray from the source and 4 the azimuth of the recording station. If the two
events have the same magnitude mb.xop(wp)= 1.
In order to calculate the magnitude M , from the surface waves, the amplitude of
the Rayleigh waves of a given frequency 0, are measured. The ratio of the Rayleigh
455
Relative excitation of seismic surface and body waves
wave amplitudes from a double couple source to those from a dilatation source with
the same mbis
X
+
- E ) [I - (1 -CR2/2B2)E]q21
lq0(2-3 sin2e)+ql sin2e+q2 sin2el
(40[3(1 -cR2/2B2) E-3 + a R 2 / a 2 ] +2q1 &(l
(3 * 3)
where T(wR,up)is the ratio of the spectral amplitudes of the doublecouple source
function Q(t) at frequencies 0, and up;T'(wR,up)is the same ratio calculated for
Q'(0.
The expression given by equation (3.3) can equally well be regarded as the ratio
of the P-wave amplitude from a dilatation source to that from a doublecouple source
with the same M,. Either way, if one is to be able to discriminate between the two
types of point source by measuring the relative amplitudes of body and surface waves
the quantity f ( ~ ~ ) / x should
~ ~ (be~significantly
~ )
greater than unity for all values of
the parameters.
Burridge, Lapwood & Knopoff (1964) have shown that the response to a doublecouple source of any orientation may be represented as a linear combination of the
responses to three basic sources; vertical dip-slip, vertical strike-slip, and dipslip on
a plane dipping at an angle of 45".
We shall therefore study only these three cases in detail.
For vertical dip-slip, f = (0, 0, l), n = (0, 1,O) and so qo = q2 = 0, q1 = sin 4,
and
We assume for the moment that T / T ' is unity. The angle 8 varies between approximately 7" and 14" for rays travelling from a shallow source, through the mantle, to
epicentral distances of 30"-90" (Carpenter 1966). We take 8 = 10" in equation (3.4),
and the result is shown in Fig. 1, for h = h' and h = 1Oh' (the elastic properties of the
half-space are chosen throughout such that 1 = p).
It can be seen that the ratio xR/xop(which in this case is independent of cop and 4)
is well above unity for most of the range of 0, h/cR. This is partly due to the fact
200.0
-
(0)
10.0
-
8.0
-
-
(b)
A. Douglas, J. A. Hudson and V. K. Kembhavi
456
that the direction 8 = 0 is a node for P-wave generation for the vertical dipslip
source and that 8 = 10" is relatively close to the node.
The deeper the earthquake (the larger h becomes) the less the amplitude of the
Rayleigh waves that it generates, whereas the body-wave amplitude is not much
affected, so xR/xop decreases, in general, as h increases. However, if we take
cR=5kms-',h'= lkm,wehdthat,eveuifh= 1 0 k m t h e v a l u e o f ~ ~ / ~ ~ ~ i ~ 2 . 5
at period of 20 seconds (oR
h/cRN 0.6). xR/xOPN 5 at the same period i f h = h' = 1 km.
Shorter periods give even better discrimination between the two events, with the
best result for h = 1Oh' at 0, h/cR N 2; i.e. around 6-s period.
For vertical strike slip, f = (1,0,0), n = (0,1,0) and so qo = q1 = 0, q2 = sin 24,
and
Again, there is a node for P-waves at 8 = 0 and the factor sin'8 in the denominators of equation (3.5) gives rise to very large values of xR/xpo (sin' 10" N 0.03).
Fig. 2 shows xR/xop as a function of 0, h/cR for h = h' and h = 10h'.
The ratio xR/xop is well above unity everywhere except in the region of the zero
which is due to the hole in the Rayleigh wave spectrum of the vertical strike slip
fault, (see Douglas et al. 1971). Discrimination between the two events is best around
W R h/CR = 3 Or for 0, h/CR less than about 0.8.
For dipslip at 45" we may take f = (0,2-*, -2-*), n = (0,2-*,2-9, so that
41 = 0,qO = -3, q 2 = -3 ~ 0 ~ and
2 4
The term in sin'8 in the denominator may be ignored as sin'8 is 0-06 at most
(8 < 14") and so the term is no more than 0.24 in magnitude; an error of one in ten
is not serious in view of the qualitative nature of these results.
r
i
50.0
1000~0
800.0
X R 6oo'o
-
-
(b)
XDP400.0
200.0
0.0
FIG.2.
1.0
2.0
3.0
w h/C,
4.0
5.0
0.0
1.0
2.0
3.0
wh/C,
4.0
xR/xoP for a vertical strike slip fault (equation (3.5)) with A = p, T = T',
0 = 10" and (a) h = h', (b) h = 1Oh'.
5.0
457
Relntive excitation of seismic surface and M y waves
Fig. 3 shows xR/xop (which is now independent of 8) as a function of 0, h/cR for
various values of 4. In this case, 8 = 0 is a direction of maximum amplitude for
P-waves from the dip-slip fault. The result of this is that, when h = lOh', f / x o p is
below unity except for small 0, hlc,. When h = h', there is still good discrimination
except in the neighbourhood of the spectral zero at 0, h/c, N 0.8.
The foregoing results show that one of the main obstacles to clear discrimination
between dilatation and double-couple sources is the occurrence of spectral zeros in
the Rayleigh wave spectrum from the latter. If the recording instrument is of sufficiently wide band that the value of 0, can be selected away from the zero value, this
can be overcome.
When the depths are equal, high frequencies give the best discrimination, but this
is not so when h/h' is larger than about 2.1, for then the expression for xR/xopdecays
exponentially as (0, h/cR)increases. For h much larger than h', the low frequencies
appear to give the best result except in the case of vertical dip-slip, when f(0,) has
a zero at 0, = 0.
The hardest case for discrimination is when the observer is at an anti-node of the
P-wave radiation pattern. Even at low frequencies, xR/xopfor a dip-slip fault at 45"
dip does not rise above about 2 when h = 1Oh'. However we can conclude on these
results alone, that in general, the mechanism of the double-couple source generates
relatively more Rayleigh waves compared with body waves than does the dilatation
source at the same depth, except in the neighbourhood of frequencies where the
Rayleigh wave spectrum corresponding to the double couple has a zero.
The difference in the mechanism is sufficient to give reasonable discrimination
between explosions and earthquakes at the same depth. However, discrimination on
this basis alone cannot be applied when the relative depth of the earthquake to the
explosion is ten or more.
1.0
0.0
1.0
2.0
.O
FIG.3. XR/,yoP for a dip slip at 45" (equation (3.6)) with h = p, T = T',0 = 10". 9 = 0, 30°,
60°, No,
and (a) h = h', (b) h = 10h'.
7
A. Douglas, J. A. Hudson and V. K. Kembhavi
458
4. The effect of surface reflections p P and SP
If the dilatation source is sufficiently shallow, the surface reflection p P will
interfere with the first arrival. In order to allow for this, xpo must be replaced by
xlP(op)
=
I
I
Q(op) lqo(2-3 sin28)+ql sin28+q, sinZe[
*
11 + A l exp [- (2ioph'/a) cos ell *
(4.1)
3
(There are in fact two possibilities; p P arrives before the first peak of
P , 0 < ( 2 0 ph'/a) cos 8 < +K, or between the first and second peaks,
+a < (20ph1/a)cos8 < 3 ~ 1 2 . We shall consider 0 < (2wph'/a) cose < 3 a / 2 and
draw qualitative conclusions without going into much detail).
The ratio xR/xlPis found by multiplying xR/xop by
[l+Alexp [-(2iuph'/a) cos811.
(4.2)
This expression is drawn in Fig. 4 for 8 = 10". For all values of 8 up to 14",A , is
within a few per cent of - 1, and so the reflection p P almost cancels P at zero depth
and reinforces it at a depth equivalent to (aph'/a) cos 8 = +a.
If h' = 1 km, cos8 = 1 (cos 14" = 0-97),a = 6 km s-l, reinforcement of P and
p P occurs around a period of 0.7 s. However, p P only affects the second swing of P
in this case and so the body-wave magnitude as measured from the seismogram trace
will not be as large as suggested. A reduction of the signal due to p P occurs at periods
greater than about 2 s.
It appears therefore that xp/xlp may be almost twice as large as xR/xOPso long as
the body wave magnitudes are measured at short enough periods for p P to reinforce
with P in the explosion signal but not (owing to h > h') in the signal from the fault.
In this case discrimination is slightly better than that predicted in the previous section.
If the depth of the fault is also shallow enough for p P and sP to interfere with the
main signal ( 2 0 ph cos 8/a < 3 ~ 1 2we
) need to use the complete expression x p instead
of xop or x:. In Fig. 5 xop/xp is drawn for the three main types of fault with h = h'.
The principal modifying effect for a fault of general orientation is p P as A2 is
approximately -0.23, whereas A l is very nearly - 1 as before. However, in the
I
I
0.5
I
I
I
1.0
I
I
1.5
I
I
2.0
uph ' / U
Fro. 4. The expression (4.2) as a function of
cup h'/a,
with h = p, 0 = 10".
Relative excitation of seismic surface and body waves
459
5.0
4.0
X:
3.0
XP
2.0
I .o
0.0
I .25
I .oo
0.75
0.50
0.0
0.5
1.5
1.0
2.0
wh/a
FIG.5. xop/xp for the three main types of fault with A = p . 8 = lo", 4 = 60".
h = h'.
cases of vertical dip-slip and vertical strike-slip, p P has a node at 8 = 0 and so the
effect of sP is as large as that of p P for 0 < 8 < 14".
For a vertical dip-slip fault, p P reinforces P at zero depth and almost concels it
out at o h cos 8/a = 4 2 , the opposite of what happens for a dilatation source. Hence
xop/xp peaks strongly near oh cos 8/a = n/2. For a vertical strike-slip fault, p P and
P cancel and reinforce in the same way as for a dilatation source. However xop/xp
peaks near oh cos8/a = 4 4 owing to the effect of sP.
The effect of p P from a dipslip fault at 45" dip is the same as from the vertical
strike slip fault and P andpP are close to an antinode in this case and so SP has
relatively little effect. This means that xop/f varies very little from unity.
It is clear that, in general, the effect of inclusion of p P , sP is to improve discrimination between faults and explosions as xop/f is greater than unity almost
everywhere for oh/u > 0.5 (i.e. periods less than 2 s for h = 1 km,a = 6 km s-').
Conclusions
It appears from this study that one of the main reasons for the larger amplitude of
surface waves relative to that of the body waves recorded from earthquakes compared
with corresponding ratio for explosions lies in the different character of the source
mechanisms. Double-couple sources, in general, generate more Rayleigh waves than
dilation sources. This may be related to the fact that all the energy of the dilation
460
A. Douglas, J. A. Hudson and V. I(. Kembhari
source goes, in the first instance, into P-waves, this giving maximum amplitude to the
first body wave arrival. The balance of P-and S-waves generated by a double-couple
source, on the other hand, appears to be a more efficient generator of Rayleigh waves.
These conclusions are based on a very simple model. Clearly they will be modified
by crust and mantle layering and by the effect of the different source time functions
[Q(t) and Q ( t ) ] . Marshall (1970) has concluded that the characters of the time
functions are such that the relative amplitudes of surface waves from earthquakes
compared with those of explosions are increased further.
The results are modified also by the difficulty of estimating depths of teleseismic
events. Surface wave amplitudes are fairly sensitive to variations in depth and,
compared with an explosion at 1 km depth, an earthquake loses its relative excess
of surface waves if it occurs below about 10 km.
Acknowledgments
Mr V. K. Kembhavi has been supported during the period of this research by a
grant from the Natural Environment Research Council.
A. Douglas:
UKAEA,
Blacknest,
Brimpton,
Reading RG7 4RS
Berkshire.
J. A. Hudson and V. K. Kembhavi:
Department of Applied Mathematics and
Theoretical Physics
University of Cambridge
Silver Street,
Cambridge.
References
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Einfluss auf das seismische Wellenspectrum. Gerl. Beitr. geophys., 71, 5-26.
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sources near a free surface, Bull. seism. SOC.Am., 54, 1889-1913.
Carpenter, E. W., 1966. A quantitative evaluation of teleseismic explosion records,
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Douglas, A., Hudson, J. A. & Kembhavi, V. K., 1971. The analysis of surface wave
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space, Bull. Seism. SOC.Am., 56, 173-183.
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