Gcophys. J. R. astr. SOC.(1969) 18, 353-370.
A Quantitative Evaluation of Seismic Signals at
Teleseismic Distances-II
Body Waves and Surface Waves from an Extended Source.
J. A. Hudson
(Received 1969 March 18)
Summary
Expressions are derived for the body wave and surface wave displacement
at epicentral distances of between 30" and 100" from an extended or moving
source. The source is assumed to lie entirely within a finite region on a
plane. Otherwise it can be quite general.
The effects of layering at the source and receiver are taken into account.
Attenuation due to linear anelasticity is allowed for by an empirical
factor. Propagation through the mantle is assumed to follow ray theory
and the sphericity of the Earth is taken into account by the use of geometrical spreading factors.
1. Introduction
Expressions for the surface waves generated by a point source in a layered halfspace have been given by both Haskell (1964) and Harkrider (1964) using essentially
the same method (i.e. the Thomson-Haskell matrix theory) but with different notations. Later on, Fuchs (1966) derived similar formulae in Harkrider's notation for
the body waves radiating into the lower half-space. These correspond to the waves
from a seismic source which travel through the mantle before being refracted back
to the surface by the velocity gradient.
A scheme by which the body wave pulse from a seismic source may be calculated, allowing for the effects of transmission through the mantle and crust, was
given by Carpenter (1966). The analysis applies to the waves recorded at epicentral
distances between 30" and 100"; i.e. waves travelling along a ray path which lies
partly in the mantle and is unaffected by the core. Kogeus (1968) applied Fuchs's
results to Carpenter's theory to allow for the effects of the layered crust. He derived
teleseismic waveforms due to an explosive source near the surface.
A method for extending these results to sources of finite extent was indicated
by Harkrider (1964) who derived expressions for the surface waves radiated from a
source consisting of a horizontal point force moving with finite speed along a line.
More realistic models of explosive and earthquake sources and their integration
into the Thomson-Haskell theory are given by Hudson (1969) (Part I of the present
Paper)Methods are, therefore, available for constructing waveforms of body waves
and surface waves at distances in the range 30"-100" from a wide range of source
models.
We shall begin by deriving expressions for the Fourier time transforms (with
transform variable o)of the body waves and surface waves from a point source of
353
354
J. A. Hudson
general type using Haskell's (1964) notation. The overlap with Haskell (1964) and
Fuchs (1966) will be minimized, but more attention will be paid to the complex
integration for the whole range (positive and negative) of w.
2. Surface waves from a point source
The problem to be considered is that of a point source in the m-th layer of an
N-layered elastic medium with an upper free surface, Each layer is homogeneous
and isotropic with Lam6 constants A,, pl and density p , ( I = 1,2, ...,N) and the
N-th layer extends downwards to infinity.
We set up cylindrical polar co-ordinates (r, (6, z) with origin on the surface
vertically above the source and the z-axis downwards into the layers. The layer
interfaces are the planes z = z1 (I = 1, 2, ..., N - 1) and the source is situated on the
planez =zm_,+hm.
Stress-motion vectors B', B", b" and b" are defined as in Hudson (1969) (Part I
of this series). B" and B"refer to P and SV-motion, 'b and b" to SH-motion. We
take the following results from Hudson.
The stress-motion vectors at the top of the lower half-space (at z = zN-,) are
given by
B'(k, n, Z N - 1, 0) = AN-i(dN- 1) ... Ai(d1) B'(k, n, 0 , (u)
+AN- 1 ( d -~I) * * * A m + 1 (dm + 1) Dm(dm-hm) S"' (2.1)
b'(k, n, Z N - ~ W)
, = aN-i(dN-1) ... q(d1) b'(k, n, 0,w)
+a~- I ( d -~1) * * * %+ 1(dm + 1) dm(dm -hm) s"',
with similar equations for Bsand b", where Al and a, are the layer matrices, d, are
the thicknesses of the layers (d, = zl --q - 1), S"" and s"" are the source vectors, and
k, n and w are the transform variables for r, 8 and t respectively (n is a non-negative
1
integer); also
Dm(z) = Am(z) E m
dm(z) = %(z)
em
where Emand emare matrices depending on the elastic constants of the m-th layer
and on k and w.
Since the surface z = 0 is free from stress,
B'(k, n, o, 0 ) =
(T)
; b'(k, n, o, w) =
(2.3)
where U:(o), U,"(o) and UrnC(o)
are unknown functions of k, n and w ,from which
the surface displacements can be calculated. Similar equations hold for Bs and
b" in terms of U,S(o), U / ( o ) , and UrnS(o).
In the half-space z > z ~ - only
~ , outgoing waves can occur. Therefore,
Seismic signals at teleseismic distances-II
355
(with similar equations B' and b') where KOC,K1', k,' are further unknown functions
of k, n and w from which the motion in the half-space can be calculated.
Substitution of equations (2.3) and (2.4) into (2.1) gives a set of four equations
in four unknowns (U;(o), Uzc(o),KO' and K,") and a set of two equations in two
unknowns (U6'(o) and k,').
The first set of equations is
J = EN-'AN-l(dN-l)
... Al(dl)
where
If we eliminate KOcand Klc, we get
where
and J , , L,,SP denote the elements of J, L and S"".
Similarly U,S(o) and U,S(o) are given by equations (2.6) with g,"' and g,"c replaced by g,"" and g,"" which, in turn, are defined in an exactly analogous way in
terms of the source vector S"'.
The equations describing the SH-motion give
where
j = eN-laN-l(dN-l) ... a,(d,)
Eliminating k,' we get
(2.9)
(2.10)
356
J. A. Hudson
The Fourier transforms in time of the surface displacements are (using Hudson
1969)
w
-
9
4,
0
9
0
)
(g,"' cos n 4 +g,"" sin n$)
=
n=O 0
-
aJ (kr) 1
ar
' F
n
- - (gccos n4 +g;" sin n$) J,(kr)l F L
r
m
q r , 4, 0 , w) =
m
2
dk
(+
(g:
cos n4 -g,"" sin n4) Jn(kr)/F
'I
(2.11)
n=O 0
- (g4"' cos n4 -g+"'
sin n4)
ar
FL
m
Gz(r,4, 0,w) =
2
dk ((gF cos nd, +g,"" sin n4) Jn(kr)/F
n=O 0
If we now write
(2.12)
and ignore terms in L ,we obtain the expressions given by Haskell (1964) which are
r
valid at large distances from the source.
The integrals over k are of the form
Z 1 -
J f ( k , o)Jn(kr)k d k
(2.13)
0
where f (k, o)as a function of k has poles at the zeros of F(k, w) or F,(k, w) and
(= v p N ) ;
branch-points at the zeros of (kz-O'/C+,'>* (= v,N) and (kz -0'/j3~')*
i.e. at k = +w/ct,, +w/BN. The layer matrices A, and a, (I = 1,2, ...,N - 1) are
~ ) vp, (= (k2- 02/Brz)*) ;
unchanged by a change in sign of v,, (= (kz - w ' / E , ~ ) or
E,S"' and other similar expressions represent the discontinuities in B" and the
other stress-motion vectors and do not contain either v,, or vpm. Therefore, f has
no branch-points at k = +w/cr,, +w/j3,, I = 1,2, ..., N - 1. Moreover, J, L, j, and
1 are even functions of k, but S"', S"",s"' and sns are odd if n is even and even if n
is odd. Therefore, f is odd if n is even and vice versa.
We can now find the parts of integrals in equation (2.11) which contribute to
the surface waves, using Lapwood's (1949) method.
If the disturbance begins at a finite point in time, the Fourier transform is analytic
if Im(o) c 0. The branch-points in the k-plane, therefore, lie off the real axis. We
assume that there are no poles on the real axis. We have defined (in Part 1) vUN
and vgNto have positive real parts and so we choose branch-cuts along Re(v,,) = 0,
Re(vpN)= 0.
We write
Jn(lcr)= +[H,,(')(kr)
+H,,(')(kr)]
= +[-exp (- inn) H,,(')(kr exp (in)) +H,,(')(kr)].
(2.14)
357
Seismic signals at teleseismic distances-I1
Using this and the symmetry properties off, we can write the integral I , as
I
00
I , =3
f ( k , w)H,")(kr) k d k ,
(2.15)
-m
the path of integration passing below the branch-point at the origin.
We now distort the path of integration around the circle at infinity in the k-plane
for Im(k) < 0, with loops around the branch-cuts and residues at the poles. The
contribution from the circle at infinity vanishes owing to the asymptotic behaviour
of H,,(').
Lapwood showed that the contribution to the surface waves comes from the
residues at the poles. Therefore, the part of equation (2.15) which represents surface
waves is
- SriCi ici f d ( i c i , w)~ n ( ' ) ( i c ~r),
where
(2.16)
f d ( r c i , w ) = lim [ ( k - ici) f ( k , w)]
k+K,
and k = K ~ ( w i) =
, 1,2, ..., are the poles offlying in the half-plane h ( k ) < 0 with
Re(V,N) > 0, Re(vsN)> 0.
We may now allow the integration path in the w-plane to approach the real
axis. The expression for the surface wave contribution remains equation (2.16)
with real o lying in (- co, co). Physically we may deduce that the poles included
in the sum are those for which k is real, positive when w > 0 and negative when
w < 0, since any other values either lead to exponentially attenuating waves which
will not be seen at large distances, or to waves propagating towards the source,
which are physically impossible. Since F and F , are even functions of k and w
we have x i ( - @ ) = -q(w). Moreover we may expect (q)' > (w/PN)' for all i so
that vaNand vBNare real and positive and the displacements attenuate with depth.
Using the asymptotic expansion of H:'), we obtain for (2.16),
(2.17)
for points at large distances from origin, where the upper or lower sign is taken for
w 3 0. This result can be applied to equations (2.11) to give the displacements at
the surface at large epicentral distances due to surface waves:
358
J. A. Hudson
where the superscript L on
of F(k, w ) only;
K:,
denotes the roots of FL(k,o)and
lci
are the roots
and
><
(putting F' = JF/Jk, F', = aFL/Jk and taking upper and lower signs for w 0).
Haskell's (1964) expressions for the surface wave displacements are correct for
o positive only.
3. Body waves from a point source
(g)
The stress-motion vectors of the P-SV-motion in the half-space are given by
B'(k, n, Z, O ) = AN(z--zN-1 ) EN
(3.1)
(using equation (2.4)) and a similar equation for B". The displacement components
are, therefore,
1
u:(z) = - - [KO' exp [-vaN(Z-zN-l)]fvbNKIC exp [-v#N(z-zN-1)]]
PN
U>(Z) =
1
-[vuN KO'
exp [-vnN(z-zN-,)]+k2 KICexp [ - v # N ( Z - z N - l ) ] ] -
PN
I
(3 * 2)
U,S and Up are given by similar expressions.
Again ignoring terms in l/r we get for the time-transform of the displacements
in the half-space (Hudson 1969)
w,4,z, w)
=
3jr
n=O 0
\
Jn(kr)
P N F(k, w,
((fine
cos n4 +f,"" sin n4) exp [ - vUN(z
-zN - 1 ) ]
Seismic signals at teleseismic distances--ll
359
On solving equations (2.5) for KOcand Kle we get
(3.5)
corresponding expressions exist for fl,,.., A,"".
The SH-motion in the half-space is given by
(3.6)
which gives
1
= -k,' exp [
U,'
- vBN(z
-2,
-
,)I,
(3.7)
PN
with a similar expression for U$(z).
The time-transform of the +component of displacement is, therefore,
((h,"' sin n$ -h,"" cos n4) exp [ - v ~ N ( z - z N -
1)])dk,
where, on solving equations (2.8) for koC,we have
the corresponding equation holds for IZ,'~.
The expressions for iir, ii, and 4, in equations (3.3) and (3.8) may be divided
into two parts. The first part consists of the terms in iir and iiz involving
exp [-vaN(z-zN-l)l
in the integrands. These components have zero curl and correspond to P-wave
transmission. The rest of the terms involve exp [ - v ~ N ( z - z N - J] in the integrands,
have zero divergence, and correspond to S-waves. We denote the two types of displacement by iip and ii' respectively; iis will be separated into an SV-component
Us" and an SH-component, GSh.
We need to evaluate integrals of the form
12 =
J
f ( k , o)exp (- va,z') J,,(kr)k dk,
(3.10)
0
where Z = Z - Z ~ - ~and f ( k , o) is an analytic function of k for l m ( o ) < 0 with
poles at the zeros of F(k, o)or FL(k,w) and branch-points at k = _+ w/uN,_+ ojp,,..
Branch-cuts are chosen along Re(vaN)= 0, Re (vgN)= 0.
It can be shown, by the arguments of Section 2, that f is an odd function of k
if n is even and an even function if n is odd. We can, therefore, write equation (3.10)
360
J. A. Hudson
in the form
m
I,
=
+
f (k, 0)exp (- v,NZ) H,'"(kr) k dk.
At large distances we replace H,")(kr)
12
by its asymptotic form to get
f f ( k , o)exp
'in[(n'2)+'1}
(2nr)f
A
(3.11)
(- vaNZ- ikr) k* dk,
(3.12)
-00
the path of integration running under the branch-point at the origin.
Lapwood's (1949) work indicates that for a transmission path running steeply
down into the half-space, the method of steepest descents gives the best approximation to the integral (3.12) at large distances.
The saddle point is given by
(3,13)
where 0 is the angle made by the downward vertical with a line from a point at the
base of the layers directly below the source (r = 0, z = z N - to the point (r, z);
tan8 = r/Z.
The solution of equation (3.13) shows the saddle-point to be at
0
k = -sin&
iw
v , ~= -cos0
EN
(3.14)
UN
(taking into account Im(co) < 0, Re(vaN)> 0).
Integrating I , by steepest descents, and assuming that there are no poles off
lying between the real axis and the steepest descents path we get,
+
where R = {r2 (Z)*}*.
( R , 0 and 4 now form a spherical polar set of co-ordinates
with origin at r = 0, Z = 0).
When evaluated in this way, the P-wave parts of equations (3.3) become
I
m
ii/(r,
4, z, co) = C
APsin8
n=O
where
A" =
uN PN
cote
F((w/EN) sin 8, W)
exp (- iwR/aN)
R
'
On using the third of equations (3.5) we find that the displacement in the P-wave
is radially outwards from the point at theLbaseLof the layers"_r = 0, i = 0) and is
361
Seismic signals at teleseismic distances-11
given by
00
n=O
(3.17)
where fi is a unit vector in the R-direction.
The S-wave part of the displacement in the (r,z) plane, (i.e. the SV-wave) is
calculated as above from equation (3.3).
where
and 6 is a unit vector in the &direction.
The SH-wave displacement can be calculated from equation (3.8).
It is
where
and 4 is a unit vector in the &direction.
We can now take the path of integration of w as near as we like to the real axis
so that w is always real.
4. Surface waves from an extended source
We now wish to consider sources whose dimensions cannot be neglected in
comparison with the wavelengths of the radiation. We shall deal only with sources
which are active on a plane surface, although the extension to non-planar and volume
sources is quite straightforward.
It will be assumed that the mechanism of the source is the same at every point.
If it is not then the expressions for the radiation may be found by superposition of
sources with different mechanisms. In order to facilitate the use of Haskell's (1964)
source vectors, we describe the source as a force system of magnitude F(r',+', z', t )
acting at points (r', +', z,,,-~ +h,+z') on a fault plane in the m-th layer at time t .
A force system (i.e. simple force, single or double couple, etc.) may be used to
describe the mechanism of the fault (Hudson 1969) and Haskell's source
vectors represent unit force, unit couple, etc.
Equation (2.18) gives expressions for the Fourier time-transforms of the surface
waves from a point source. If we use source vectors representing the unit force
system, then the surface waves from an element of the fault at the point
(r', 4',ZnI--l+hfl+Z')
362
J. A. Hudson
are given by equation (2.18) multiplied by .F(r',
4', z', a), the time-transform of
9;
e.g.
%lg(rl,41, 0
= F C Ail
i
I
, ~ )
m
lci
C [grlnC(q,
w ) cos n+l +grlnS(rci,w ) inn$^] exp (inn/2).
n=O
The subscript 1 indicates that the co-ordinates (rl,+,) in any plane of constant z
are measured from an origin at r = r', 4 = 4'. We define the lines 4 = 0 and 4, = 0
to be parallel. Further h, is replaced by h,+z' to form g,,"' and g,,"".
We now integrate over the fault plane I; to find the total surface wave contribution from the fault. If the linear extent of the fault is small compared with distance
h, to lie on
of the receiver from the fault and if we take the point r = 0, z = 2-,
the fault, then r f / r and zf/r are small quantities.
The r,-components of displacement in equation (4.1) are approximately radial
from the axis r = 0 for all rl, i.e. in the r-direction. Therefore the integral over Z
is a scalar sum;
+
00r
+
f
1
gr,"sA , , B sin n#l do exp (inn/2).
z
(4.2)
Neglecting small terms, we may replace 4, by 4 in equation (4.2), and replace
rl by r except in the exponential term in Ail, where we put
Y1
=r-
s.r
r
(4.3)
where r is the vector position of the point (r,Cp) in a horizontal plane and s is the
position of an arbitrary point on the fault plane relative to an origin at r = 0,
z =~,-1+h,.
In order to eliminate as many unnecessary parameters as possible, we choose the
plane 4 = 0 to be parallel to the intersection of the fault plane with the horizontal.
We also define the point s on the fault by polar co-ordinates (s, 8') relative to an
+h,, with the line 8' = 0 lying in the 4 = 0 plane (and
origin at r = 0, z = 2-,
therefore horizontal) and increasing in a positive screw direction along the normal
n to the fault.
Equation (4.3) is now
ri = r-Ns,
where
(4.4)
+
N = cos 8' cos 4 n3 sin 0' sin 4
and n3 is the direction cosine of n relative to the z-axis.
In many cases we will be able to separate the function P into a product of a
function of time and a function of position. In the general case, when this is not
possible, we make the separation
F(r',
4', z', w ) = Q(o)F(s, e', w )
(4 * 5 )
where Q is an amplitude factor with the dimensions of the force system and F is
primarily a spatial variation function with magnitude of the order of unity. In the
special case referred to above, F does not depend on w.
363
Seismic signals at teleseismic distancei-II
m
Q rci A; nC= 0 [G,"c(rcf, o)cos n4 + G,""(rc,w ) sin n+]exp (inn/2),
i
=C
(4.6)
where (analogously to equations ( 2 . 5 ) and (2.7))
4
G,"E(k?
=
C [(J12-J*2)(E3i-L41)-(J32-J42)(Lli-L2i)ISinC
i=l
(4.7)
with a similar expression for G,""(k, 0);
-
... Dnl
E = EN-'AN-l(dN-l)
and
1
a, =
1
w,0)exp ( i K i 1 ~ x 1do.
~ , , , ( d ,- h, -zi> ~ ( s ,
I:
(4 * 8)
The elements of Drn(dm-hm)are linear combinations of cosh and sinh with arguments varn(d,,,-h,) and vsm(d,-hm). Therefore, the elements of '&,are exactly the
same except that
isfeplaced by
Ca(k, 0,d m - h m )
Sa(k, 0,d m -hm)
I
= t [ e x ~[vm(dm-hm)I
Ea-(k, o ) + e x ~[-vmt(dm-hm)I
2n
where
E#*
do'
(k, o)=
0
fF
~,+(k,
0)]9
exp [is(Nkk iv,, n2 sin O')] sds
(4 * 9)
(4.10)
0
assuming F is zero outside the source region; iz2 is the direction cosine of n relative
to the axis z = 0, = n/2. The vector n may be written as (0, n2, n3) relative to
Cartesian axes
x1 = r cos 4
x, = r sin+
(4.11)
+
x3 = z.
These are the axes used by Haskell (1964) to define his source vectors.
Similarly
364
J. A. .Hudsad
We can now evaluate the radial component of the surface wave contribution
from a source of fairly general type. The only difference between equation (4.6)
and the result for a point source lies in the components of
The other two components of surface wave displacement are
a,.
=
c
m
QK?
A:
c
[ c 4 n c ( K i ~ ,w )
n=O
sin mj - G+.~S(K?,
0 )cos
n4l exp (inn/2), J
where G,"", G,"" are calculated in the same way as g,"", g,"s (equation (2.7)) except
replaces D,; G+.""
and G," are calculated in the same way as g4" and g4"*
that
except that a, replaces d,, where in a,,
n,
5. Body waves from a moving fault
The Fourier time-transform of the P-wave from an element of the source at the
point (r', 9',z,- +h,+z') is given by equation (3.17)¶
,
iP(r1,
41, z, 4
1
sin 91 exp (inn/2) (5.1)
where the subscript 1 indicates that rl, R , , el and 4,, are measured relative to an
origin at r = r', 4 = I$', z = Z N - 1 , and h, is replaced by h,+z' in f,,"" and frlM;
R, is the point (rl,
Z-zN-1)
and R, = RJR,.
We now integrate over the fault plane and assume that the linear dimensions of
the fault are small compared with the distance R to the receiver from the base of
the lowest layer below the origin of faulting. So we put 9, = t$y = 8 and Rl = R
everywhere except for the exponential term in Alpyin which we put
R 1 = R-
s,.R
R y
where R is the vector representing the point (r, 9, z-z,- ,) and s, is the projection
onto the horizontal of the vector position s of a point on the fault. In terms of
previously defined angles,
R, = R-Nssintl,
where N is defined in equation (4.4).
-
( 5 3)
365
Seismic signals at teleseismic distances-I1
Hence, we have for the P-wave from the extended fault,
up(r,(6, z, w)
m
=
2
OAPR
n=O
+ FF
(z
sin 0, w) sin n$] exp (inn/2), (5.4)
where F,""(k, w) and F,""(k, w) are calculated exactly as in equations (3.5) for
f,""(k,w) and,f,""(k, w) except that L(k, w) is replaced by L(k, w), which is defined
in equation (4.8). In the evaluation of equation (5.4), k is put equal to w sin 8/aN
throughout.
Corresponding to equation (5.4) we have
iis"(r,(6, z, w) =
m
C
n=O
QAsv6
+H F: (
)
sin O, w sin n 4 ] exp (inn/2), (5 . 5 )
where H,""(k,w) and H,""(k,w) are defined in the same way as h,""(k,w) and
h,""(k,w) in equation (3.5) except that L is replaced by L.
Similarly
m
iish(r,(6, z, w ) =
C QASh4
n=O
-HT
(i
sin 6, w) cos n(6] exp (inn/2), (5.6)
where H,"C(k,w) and H,""(k,w) are defined as in equation (3.9) except that 1 is
replaced by I, which in turn, is defined in terms of a, and a is defined in Section 4.
6. Surface wave disturbance at the receiver
Now we have expressions for the radiation from the source we can introduce
the effects of the transmission path to find the response of a seismograph at a distant
point on the Earth's surface.
The surface waves emanating from the source are given by
U,4
=
UZ4 =
2a F/'((6, w) r - 3 exp (- i~~ r
in/4)
C F2'(4, w) r - 3 exp (- ilci r & in/4)
I
Go4 =
B
F,qi((67w) r-* exp (- irciLr & in/4),
3
366
J. A. Hudson
where the upper or lower signs are taken for w
[G,"C(q,
F14i= -
2n +
0)cos
>< 0, and (equations (4.5) and (4.13))
n 4 + G F ( q , w) sin n4] exp (inrr/2) I
Qi
(
,
i
) 5
F'(IC', w ) n = o
[G+~c(IC?,
w) sin n4 - G ~ " s ( I C ~ ~w)
, cos n4l exp (inn/2).
I
(For a point source, Q is the magnitude of the source and the G,""etc., are replaced
by g,"', etc.)
To allow for the absorption of energy by anelastic processes in the Earth we
multiply the displacements by empirical factors
exp (-rlwl/2Q'Ui)
and exp (-rlwl/2QL' U L f ) ,
where Q'(w), QLi(w) are the attenuation factors of the corresponding modes of
surface waves and U'(w), U',(w) are the group velocities.
Corresponding to attenuation there is dispersion (Futterman 1962) and so there
will be a change in the values taken by IC, and I C due
~
to anelasticity.
We have assumed, in applying the above analysis to the Earth, that all relevant
wavelengths are small compared with the radius of the Earth. For waves travelling
large distances, however, we have to replace the geometrical spreading factor r-*
by that appropriate to a sphere, (a sin A)-* where a is the radius of the Earth and
A the angular distance travelled by the wave.
In its path, the surface wave train may cross continental boundaries or other
changes in the vertical structure of the crust and upper mantle. We may allow for
this in principle by multiplying the Fourier transforms of the displacements by
transmission factors 7','(0), T,'(w), T+'(o)and by replacing I C ~I, C by
~
the wave
:~
to the structure at the receiver.
numbers K:, I Cappropriate
Finally we multiply the displacements by S,(w) or S,(w) to allow for the response
of the recording instrument; S,(w) is the response of the horizontal instrument and
S,(o) that of the vertical.
We have
12," =
CS,(o) T,'(o)F,"'(+, w)(a sin A)-* exp
I
I22 = C S,(w)
Tzi(w)Fzqi(4,w)(u sin A)-* exp
1
fib*
= C S,(w)
i
4
T+'(w)F+qi(+, w)(a sin A)-*
exp (-iK:Lr+
with r = aA.
in
in -
rbl
i ,
2Q0'Uo
)
(6.3)
367
Seismic signals at teleseismic distances-11
7. Body waves displacements at the reciever
The formulae for the body waves travelling into the lower half-space may be written
in the form
iip = RFP(8,4, w) eXp (- iWR/CIN)
R
I
where (equations (5.4), (5.5), and (5.6))
w sin8
+HT (
7
, w) sin n4] exp (inn/2)
(For a point source, Q alone represents the magnitude and F,"C, H,"C etc., are
replaced by f,"',h,"", etc.)
We consider body waves which travel through the mantle without being affected
by the core before arriving at the crust below the receiver.
Equations (7.1) represent waves travelling into a homogeneous half-space.
Since we have approximated for large distances, the wave-fronts are spheres centred
on the point R = 0 at the base of the layers vertically below the source. (There is
an initial phase, given by the argument of FP, F " , or FShwhich corresponds to a
time-delay due to travel through the layers and which depends on 8 and 4.)
We may use this approximation to describe body waves travelling through the
mantle if the separation of the rays at the base of the crust is small compared with
the epicentral distance of the receiver. Carpenter (1966) showed that rays travelling
epicentral distances of between 30" and 95" have angles of incidence between 7"
and 14" at the surface. This means a separation of a few kilometres between the
rays at the base of the crust which may be neglected.
368
J. A. Hudson
In general, this approximation is valid if the variation of properties of the mantle
is small within a wavelength. If this is so, we may also apply the results of ray
theory to find the amplitudes of the waves arriving at the base of the layers below
the receiver. Following Carpenter (1966) we get the P-wave amplitude to be AoP
where
lAoplZ= (Mp)' lFP(ip,rb,w>I2
(7.3)
where
sin i
dip
p 2
'NpN
-C
I a
,
' sinA:osi:
~
~ IdA ~
~
~
1
and mM0, p M o ,are the velocity and density of the material immediately below the layers
at the receiver, a, is the distance from the base of the layers below the receiver to the
centre of the earth, ip is the angle made with the downward vertical by the ray as it
leaves the base of the layers below the source and ipo is the corresponding ray angle
at the base of the layers below the receiver; A is the angular separation of source and
receiver (assuming this to be approximately equal to the angular separation of the
points where the ray path crosses the bases of both sets of layers).
The angles ip and ipocan be found from the formulae (Bullen 1963)
where a, is the distance from the base of the layers below the source to the centre
of the Earth and Tpis the travel time of P-waves from source to receiver.
At the base of the layers below the receiver the wave-front may be regarded as
plane, with constant amplitude along the front. The phase of the wave may be
calculated from the travel-time T,' between points on the ray at the base of each
set of layers. The P-wave may be represented there as
lip = ipoFP(ip,4, o)M P
exp(-i[oT,'+,jxsini,O-(z-z~-,)cosi:}])
0
,
(7.5)
UM
where (x, y , z ) are Cartesian co-ordinates, origin at the receiver on the surface with z
vertically downwards and with the x - z plane containing the ray path; z =
is the depth of the base of the layers below the receiver; ipo is a unit vector in the
direction of propagation of the wave (ipo = (sin ipo,0 , - cos ipo) in the ( x , y , z )
co-ordinates).
Similarly, we may find expressions for the displacements of the shear waves.
us"= is>Fs"(is,cj,o)M"exp (-i
-
[wT,'+ ~ 0
{ x s i n i ~ - ( z - z & _ , ) c o s i > } ] (7.6)
)
PM
where
and where the subscript s refers to shear waves as opposed to dilatational waves and
the superscript 0 refers to the properties of the material below the receiver. iiShis
given by the same expression with :si Fsh replacing is> F " ; is: and is: are unit
vectors in the directions of polarisation of the corresponding waves.
We assume the crust below the receiver may be represented by M - 1 homogeneous layers overlying a uniform half-space. The effect of this crustal layering on
the displacements can be allowed for by Haskell's (1962) method. Changing to
369
Seismic signals at teleseismic distances-I1
Haskell's later (1964) notation (i.e. the notation used here) we find that the horizontal and vertical components of displacement at the surface are given by multiplying the P-wave displacement at the base of the layers (z = zg- ') by
and
respectively, where the Jij0 are the components of
the superscripts 0 on the layer matrices Al0(dlo)and on EMoimply that they refer
to the crust below the receiver.
We can allow for absorption by multiplying by the empirical factor
(Carpenter 1966) where Q, is the attenuation factor for P-waves, which varies with
depth, and the integration is taken along the ray path from source to receiver.
Finally we allow for the response of the recording instruments through the factors
&(a) and &(a). The recorded displacements are (at x = 0 and referred tc c, y , z )
co-ordinates)
EXp(o)= &(w) CXp(w)M P Fp(ip,4, w) exp
(7.10)
UZp(o)= S,(O) Czp(w)M P FP(ip,4, co) exp
The corresponding result for the incoming SV-wave is
where Qs is the attenuation factor for shear waves. The factors compensating for
the layers at the receiver are
C,"(w) =
i
(7.12)
C,s(O) =
The incoming SH-waves give rise to displacements
370
J. A. Hudson
where
C,”(w) =
2PM0
FLo((o/pMo)
sin i:, o)
and FLo(k,o)is formed for the crust at the receiver in the same way as F,(k, w )
for the crust at the source.
8. Conclusions
The results obtained above provide a general scheme by which the displacements
due to surface waves and to body waves at epicentral distances between 30” and
100” from a source of fairly general nature can be calculated.
The theory suffers from a lack of precise knowledge about the non-linear and
anelastic behaviour at an explosive source and the effect of the cavity; also about
the nature of an earthquake source. Furthermore, the approximation of using
ray theory through the mantle may not be sufficiently accurate, particularly in respect
of the generation of waves of the opposite type by P or S-waves.
The transmission coefficients for surface waves passing mountain ranges or
continental boundaries are completely unknown, and any form of diffraction by
lateral inhomogeneities in the Earth have been neglected. So also has the scattering
effect on body waves of irregularities in the crust near the receiver.
Acknowledgment
This work is being undertaken in collaboration with the seismological group
at U.K.A.E.A. Blacknest. I am indebted, in particular, to Dr H. I. S. Thirlaway,
who suggested the problem, and to Mr A. Douglas and Mr P. D. Marshall for
discussions and criticism.
Department of Applied Mathematics and Theoretical Physics,
Silver Street,
Cambridge.
References
Bullen, K. E., 1963. Introduction to the theory of seismology, 3rd edn, Cambridge
University Press.
Carpenter, E. W., 1966. Proc. R . Soc., A.290, 396-407.
Fuchs, K., 1966. Bull. seis. SOC.Am., 56, 75-108.
Futterman, W. I., 1962. J. geophys. Res., 67, 5279-5291.
Harkrider, D. G., 1964. Bull. seis. SOC.Am., 54, 627-679.
Haskell, N. A., 1962. J. geophys. Res., 54, 4751-4767.
Haskell, N. A., 1964. Bull. sei8. SOC.Am., 54, 377-394.
Hudson, J. A., 1969. Geophys. J. R . astr. SOC.,18, 233-249.
Kogeus, K., 1968. Bull. seis. SOC.Am., 58, 663-680.
Lapwood, E. R., 1949. Phil. Trans. R. SOC.,
A.242, 63-100.