TOURNAT.OF GEOPHYSICALRESEARCH
VOL. 69, NO. 12
JUNV. 15, 1964
Radiation Patterns of Seismic Surface Waves from Buried Dipolar
Point Sources in a Flat
ARi BEN-MENAHEM
Stratified
Earth
AND DAVID G. HARKRIDER
SeismologicalLaboratory
California Institute of Technology, Pasadena
Abstract. Explicit compact expressionswere obtained for the far displacement field of
Rayleigh and Love waves generated by force configurations which served to simulate sheartype faults with arbitrary dip and slip. The medium transfer functions for dipolar sources
were computed for a Gutenberg fiat continental earth model with 23 layers. These were
then used to obtain universal radiation pattern charts for couple- and double-couple-type
sourcesat various depths over the period range 50 to 350 sec. It was demonstrated by means
of few typical examples that the radiation patterns of Rayleigh waves may depend strongly
on the depth of the source,and unlike the fundamental Love mode may be rather sensitive
to small variations in frequency. For a given source and frequency the radiation pattern
may differ considerably from one mode to another.
INTRODUCTION
terns for Rayleigh waves in a half-spacedue to
The inverse problem of reconstructingthe moving faults. Harkrider [1964b] calculatedthe
responseof a multilayered half-space to Love
seismicsourcefrom amplitude measurementsat
and
Rayleigh waves due to buried horizontal
stations encircling the epicenter was hitherto
mainly restricted to first-motion studies.In spite and vertical point forces. Haskell [1963] calculated radiation patterns for Rayleigh waves
of the fact that certain amounts of informagenerated
by dipolar point sourcesat depth in a
tion could be obtained from a tiny portion of
homogeneous
isotropic half-space.The present
the seismogram,seismologists
soonrealizedthat
paper
supplies
the generaltheory of Rayleigh
the complexity of earthquake origins in space
and time could not be adequately understood and Love wave radiation from dipolar point
and uniquely determinedunlessways and means sourceswith arbitrary elementsin a multilaywere found to extract the source parameters ered earth. The theow is then applied to a
continental Gutenberg earth model.
from the main body of the seismogram.
The advent of long-periodseismographsand
LIST OF SYMBOLS
high-speedcomputersopenednew vistas in this
field. Analysis of surface waves and free oscil- a0 Coefficientin the expressionfor the reduced
lations improved our knowledge and supplied
displacements(Table 3).
us with additional
tools for future
research.
The present availability of long-period records from the archives of the USCGS worldwide net of standardized stations will undoubt-
edly put source mechanism studies on a new
level. To prepare the groundsfor studiesof this
kind, it is necessaryto know first the theoretical
radiation pattern of surfacewavesfrom various
sources.Yanovskaya [1958] calculatedthe responseof a layer over a half-spaceto Love and
(•R•, (•
Rayleigh and Love matrices of the
ruth layer.
AR•, A•
The Rayleigh and Love product
matricesof m ]ayers.
A•, A• The Rayleigh and Love amplitude
factors.
A (h) Rayleigh wave ellipticity function.
b0, b• Coefficientsin the expressionfor the
reduceddisplacements(Table 3).
B(h, co) Rayleigh wave first dipolaf transfer
function.
Rayleigh waves due to singlets and couples.
Ben-Menahem[1961] calculatedradiationpat- c(co)Phasevelocity.
• Contribution 1237,Division of GeologicalSciences, California Institute of Technology, Pasa-
c•, c• Rayleigh and Love phasevelocities.
C(h, co)Rayleigh wave seconddipolar transfer
function.
dena.
26O5
2606
BEN-MENAHEM
AND
ItARKRIDER
d• Radiationpattern coefficients
(Table 1).
/•0 Shearvelocity at depth h -- 0.
e•, e•, es Unit vectorsat the sourcealong the /•s Shear velocity in the sourcelayer.
strike, dip, and the vertical directionsrespec- y(w) Absorptioncoefficientfunction for surface
tively.
er, e0, e, Unit vectorsalongthe radial, azimuthal
and vertical axesat the recordingstation.
(F, F e, F,) A three vectorrepresentation
of the
total far-displacement
field at the recording
station.
G(h, ,o) Love wave dipolartransferfunction.
h Sourcedepth.
Ho(•)(k,,r), H• ('.)(k,,r)Hankel functions.
kn Wave number of nth mode.
k•, kL Rayleigh and Love wave numbers.
waves.
e0Rayleigh wave surfaceelliptictry.
• Rigidity.
•s Rigidity of the sourcelayer.
p Density.
o'Rs(h)Depth-dependentfactor of normal stress
associatedwith the Rayleigh waves.
a0 Potssonratio at depth h -- 0.
•'R,•(h),•'zs(h) Depth-dependentfactorsof tangential stressesassociatedwith the Rayleigh
and Love waves.
L Conversion factor.
qbR,
qb•Rayleighand Love spectralspatialphases.
mR(O), mL(O) Rayleigh and Love radiation x(•) The radiation pattern function.
pattern amplitudes.
w Angular frequency.
M Moment of couple.
Other symbolsdefinedin Figure 1.
n Normal vector to plane of motion.
COORDINATE
SYSTEM•NOTATION,AND
No(h, ,o) Love wave singlettransferfunction.
SIGN CONVENTION
Nr•(h, ,o) Rayleigh wave first singlet transfer
function.
We consider a shear-type fault with arbiNrz(h, w) Rayleighwave secondsinglettransfer trary dip and slip in a layeredearth. We replace
function.
the physical fault by an equivalent force sysR(co)The time transformeddisplacementvector tem. This force system is generated from a
in plane of motion.
single force, which is directed along the fault's
t Time.
motion and has a magnitudeproportionalto the
T Period of a spectralcomponent.
amount of displacementin the plane of motion.
can furthermore assume, without loss of
•s(h)
9s(h)tbs(h)
Itaskell's
plane
wave
par- We
•o ' 90 ' •o
generality, that the time dependenceof the
force is that of the Dirac delta function.
ticle velocity (or displacement)ratios.
•0' = --i•0.
The geometryof the situation is displayedin
U Group velocity.
Figure 1. FollowingHaskell [1963]we choosethe
U(co)The time transformedvectordisplacement source coordinate system to be cartesian with
field.
the x• axis in the strike direction, the x, axis in
U(co) Complexconjugateof U(o•).
the dip direction,and the x• axis in the upward
U s, U ½, U " The time transformed vector dis- vertical direction. The convention is followed
placementfields due to singlet, couple,and that all three force componentsare positive on
double-couple
sources,respectively.
the hanging-wallsideof a reverse
left-lateralfault.
U,, U e, U, Components of time transformed Hence the vectorR(•0) will be representedby
displacementfield in cylindrical coordinates.
_
]U(r, O,w)] Spectralamplitudeof groundmotion
at the recordingstation.
U*(r, O,w) The reduceddisplacement.
]U*(r, O,oo)]The reducedamplitude.
q- sin h cos/te2q- sin }, sin/tea}
(1)
Likewise,the normal is given by
U0 Coefficientin the expressionfor the reduced
n = In[ {--sin /teaq- cos/tea}
(2)
displacements(Table 3).
whereO<
/t<•randO<
h<
2•r.
z Depth of receiver(station).
a Compressional
velocity.
SINGLE FORCE IN A MULTILAYERED MEDIUM
a0 Compressionalvelocity at depth h = 0.
as Compressional
velocity in the sourcelayer.
It hasbeenshownby Ben-Menahemand ToksSz
/? Shear velocity.
[1963]that the displacement
field of an arbitrary
RADIATION
PATTERNS
OF SEISMIC
SURFACE WAVES
2607
Fig. 1. Geometryof sourceelementsand relative positionof stationon the free surface.
force system in a multilayered medium can be
generatedfrom the responseof the medium to
three mutual perpendicularunit forces.
The position of the recordingstation is given
by (0, r, z - 0) in the sourcecylindricalsystem
(see Figure 1). 0 is measured positively anticlockwise from the positive strike direction
(x• = 0) when viewed from abovethe half-space
downward. r is measured positively outward
from
the source. Thus we can write
the far-
displacementfield U s due to a single force
at the source as
= (R.
+ (R.
+ (R.
sin 0 sink cos•) + iNr• sink sin $]
(S)
In theseexpressions
the factor(2wr)-•'e
has been suppressed.All other constants are
incorporatedin the definitionof N•, N,r, and
N•,. The symbolNii whichappearsin (8) and
(9) standsfor the componentof the amplitude
response
of the mediumin the i direction(horizontal or vertical) due to a unit force acting
along the j direction(horizontalor vertical).
Thus, for example,we write Nr, for the horizontal responsedue to a vertical force. These
(3) functions are obtained directly from the exact
where
solutionto the boundaryvalue problem.They
depend on the frequency,the depth of the
F• -- (sinOe'3'/4N,,cos9e-"/•N,, O)
(4) source,andthe constants
of the layeredmedium.
N• is the Love wave responseof the mediumto
F, = (cos
a horizontal force. The factor (k,) -•' which
sin 9e-•/•N•, e•/•N•)
(5) arisesfrom the asymptoticexpansionapproximation of Ho.•(")(k,,r)is absorbedin
• • (cos
Comparingequations3 to 9 with the solutions
given
by Harkrider [1964a] for vertical and
sin 9e-•/•N•, e-•/•N•)
(6)
horizontalforceat depth in a multilayeredhalfUsing •he representationof R •s given in (1) space, we can expressthe functions N, at
we ob•Mn the f•r-displ•cemen• field in •erms of Z-- 0as
•he elementsof •he fault (k, •) •nd •he •zimuth
(10)
'
•ngle 9.
N•(h) -- --[ao*/•2o]N•,(h)- ,oN,•(h) (11)
U•s • [R[
- cos• s• k cos•)
(•)
• • -•/2
N•(h) : [•2s(h)/•2o]A•
N,•(h) = eoN,,,(h)
(12)
(13)
2608
BEN-MENAItEM
N•(h)- [gs(h)/9o]ALkL
-•/"
AND HARKRIDER
(14)
terms of higher order than r-• "-. The vector n is
a unit vectorwith dimensions
of length.
The amplitudefactorsAr• and AR are functions
The double-couple
sourceis obtainedby the
of frequencyand the elasticpropertiesof the superposition
of twoequalcouples
at rightangles
multilayeredarray.They are independent
of the sothat the total momentof the systemis zero.
sourcetype and depth. This is also true for the
To obtain the far-displacementfield for this
elliptictry.The Rayleighwaveamplitudefactor systemwe add to the operationalrepresentation
AR alsooccursin the solutionto an explosive of U c asgivenby (22) the displacement
fielddue
sourceat depth. The propertiesof A• and
to a secondcouplewhich is formed by the
aregivenin moredetailby Harkrider[1964a,b]. interchangeof 1t with n. Hence
The depth-dependent
factor in equations10
to 14 canbe representedin terms of the Thomson-
U•)c= --(n.grad)US(11)
Haskelllayermatrices[Haskell,1953]as
-- ]•-grad
}U
s(R[n) (23)
[fta*(h)/tbo]
-- [A•s(h)],,.*-- eo[A•s(h)]•x
(15)
=
Performing
the operations
indicatedby (23) we
+
obtain general formulas for the far field of Love
[0•(h)/00]: [ALs(A)]11
=
...
Azs(h) = az s(h)azs-• --- az•
(17) and Rayleigh waves.The generalform of this
(is)
(19)
field may be written as
U = ]R[ In]kne-'S'/4N(h)x(O) (24)
•nd a•,, az, •re the R•yleigh•nd Lovem•trices wherek,is eitherka or kz,,N(h) is eitherN•(h)
or N,.•(h),and X(0) is the complexfunction
of the ruth l•yer. The sourcel•yer is indicated
by m = S •nd the surface
l•yer by m = 1. The X(0) -- doq- i(dxsin 0 q- d2cos0)
m•tricesa•s(h) and a•(h) •re layer m•trices
q- dasin 20 q- d4cos20
(25)
for a subl•yerin the sourcel•yer, the thickness
of which is equM to the penetrationof the The coefficientsd• are given in Table 1. k is
sourcedepth h •to the sourcel•yer. From measuredcounterclockwise
from the positive
equations10 to 14 we seetMt
strike direction,and /5 is measureddownward
from the negativedip direction.The dimension-
lessentitiesA, B, C, and G are givenby
for •ll sourcedepths.Since,by definition,
ao • a•(0) •nd •o • •(0), it followstMt for
surfacesource,h = 0,
=
GENERATIONOF DIPOLAR SOURCE
The coupledisplacements
areobtained
by the
applicationof a differentialoperatorto the displacement
vectorU s dueto a singleforce.Thus
a(a):
=
c(a):
+
•,,(a)/•,(a)
(,)
oa
oa
6(a)- 0a /
The commonphaseconstante-•s' t•, appea•g
Uc-- --(n'grad)U
s= sin0sin
-- cos
$• U + O(r
-a/a) (22)
in (24) has been chosen so as to render the
functionsA, B, C, and G non-negative
at h = 0.
The signsin thesefunctionsare adjusied• such
a way that h is increas•gpositivelydownward.
The complexfunction x(O) definesiwo real
We mustchoose
herethe negativesignfor the functions of the real variable 0. Its modulus
[x(0)[ givesthe amphtuderadiationpattern,
source,the stationremainingfixed.Sincewe are andits argument
argX(0) givesthespatialphase
gradient becausethe derivative is taken at the
interestedonlyin the far field,we havediscarded [Ben-Menah• and ToksOz,1963]of the source.
RADIATION
PATTERNS
TABLE
1.
OF SEISMIC
Radiation
Pattern
SURFACE
d0
d•
d•
d,
d•
Love
2609
Coefficients
Couple
Coefficient
WAVES
Double Couple
Rayleigh
--• cos X sin i•
• sin X sin 21•B(h)
cosX cos i•G(h) sin X[1 - C(h) cos' •]
- sin X cos• i•G(h) cos X cos i•[1 - C(h)]
• sin X sin 2 •
• cosX sin i•A(h)
• cosX sin i•
-•
sin k sin 2 i•A(h)
Love
Rayleigh
0
cos X cos i•G(h)
- sin X cos21•G(h)
• sin X sin 2 •;
cos X sin •
• sin X sin 2•B(h)
- sin X cos 2 i•C(h)
- cos X cos i•C(h)
cosX sin i•A(h)
-•
sin k sin 2 i•A(h)
From equations10 to 19 and the definition of For a surface source, h -- 0, the stressratios
the elements of the layer matrices [Haskell, defined by equations 34 to 36 vanish, and we
1953], the derivatives appearing in the expres- obtain
sionsB(h), C(h), and G(h) can be given as
G(h)=
_1{[Ars(h)]21*}/•
9s(h)]
(30)
c(o) = 6(o) -- o
/zs
(39)
L •)o
A(h) -- --[as*(h)/•
(h)]
(31)
CHARACTER OF THE TRANSFER FUNCTIONS OF
THE MEDIUM
Values of the exact expressionsfor N0,
N,, A, B, C, and G were computedfor a realistic
ß
{[AR•(h)]a•.*-- eo[AR•(h)]a•} (32)
continental
earth
model. Results were obtained
for the fundamental Rayleigh mode and for the
fundamental
and
first
Love
modes
over
the
C(h)
-----JARs(h)]42
+ eø[AR$(h)]41*
(33)period range 50 to 450 sec. Sourcedepths were
]
taken
at 5-kin intervals
between
0 and 40 km
physicalmeaningof stressratios at depth h'
and 33-km intervals between 66 and 627 kin,
27 points in all. The input layering constants
and the output phase and group velocities are
It can be shown from (31) that A(h) is the
ellipticity that a receiver at depth z -- h would
measurefor all sourcedepths.
The transfer functions are shown in Figures
2 to 20. Each function has been plotted both as
a function of the depth of the source and the
frequency.Note the slow variation of G(h) with
frequency for sourcesbelow the low-velocity
channel (Figures 3 and 4). Note also the effect
of the M discontinuityon the profile of G(h) as
compared with an analogous variation of
the shear velocity function (Figure 5). While
Nr,(h, •o) follows the behavior of No(h, to) in
both variables (compare Figures 2 and 8 and
where the functions
inside the braces have the
shown in Table 2.
Figures3 and 9), the functionN,,(h, o•) differs
A(h) = e(z = h)
A_(O)= e(0) = eo (37)
from it considerably, mainly because of its
Sincethe ellipticity is independentof the source spectral nodes, which depend strongly on the
depth of the source(Figures 6 and 7). Another
depth, we can write
interestingresult is the slow variation of B(h)
with period for sourcedepths up to 300 km
=
(38)
(Figure 12).
=
2610
BEN-MENAHEM
AND tIARKRIDER
TABLE 2. Constants
ofa Gutenberg
FlatContinental
EarthModelandCalculated
Velocities for Three Surface Wave Modes
•=
•d=
D
23 DeLl=
l DKS=
ALPtiA
6ø0300
6.7000
?.9600
22.00
15.00
13.00
25.00
50.00
75.00
50.00
I00.00
100.00
100.00
100.00
100.00
I00.00
100.00
150.00
200.00
200.00
200.00
200.00
200.00
200.00
200.00
200.00
7.8500
7.8500
8.0000
8 2000
8.4000
9 0000
9.6300
10.1700
10.5850
I0.9550
11.2750
/1.4600
11.7550
12.0200
12.2800
12.5400
12.8000
13.0200
13.2400
13.4800
RAYLEIGH WAVE SPEC[RAL VALUES FOR
PERIflU
0.4295œ
0.3970b
0.3689E
0.3444E
0.3228E
0.3032E
0.2852E
0.2683E
0.25201
0.2359E
0./854E
0.1660E
0.1438E
0.11631
0.7673E
0.3770E
0.2834E
0.2392•
0.2073E
0.1791E
0.1493E
0,/094k
C
03
03
03
03
05
03
03
03
03
03
03
03
03
03
02
02
02
02
02
02
02
02
0.5600E
O.b5OOE
0.5400E
0.5300E
0.5200E
O.5100E
O.5000E
0.4900E
0.4800E
0.4700E
0.4400E
0.4300E
0.4200E
0.4100E
O.4000E
O.390UE
0.5800œ
0.3700E
0.3600E
0.3500E
0.3400E
0.3300E
LOVE
U
01
01
01
01
01
01
Ol
01
Ol
01
01
Ol
01
0l
01
Ol
Ol
01
0l
Ol
01
01
0.4598E
0.4438E
O.4280E
0.4130E
0.399lE
0.3869E
0.3765E
0.3682E
0.3620E
0.3580E
0.3580E
U.3620E
0.3681E
0.3762E
0.3857E
0.3695E
0.3356E
0.3128E
0.2997E
0.2965E
0.3005E
0.3103E
01
01
Ol
01
01
01
Ol
01
0l
01
01
Ol
01
01
Ol
Ol
0l
01
0l
01
Ol
01
0.0000010
FK=
0.00010
1.0 CPER[=
1.00100
FKD=
BE[A
3.5300
3.8000
4.6000
RHO
2.1800
3.0000
3.3700
MU
0.3464
0.4332
0.7131
4.bOO0
4.4100
4.4100
4.5000
4.6000
4.9500
5.3100
5.6300
5.9150
6.1400
6.2850
6.3840
6.5000
6.6100
6.7400
6.8500
6.9600
7.0000
7.1000
7,2000
3.3900
3.4200
3.4500
3.4700
3.•000
3.6300
3.8900
4./300
4.3300
4.4900
4.6000
4.6900
4.8000
4.9100
5.0300
5./300
5.2400
5.3400
5.4400
5.5400
0.6865
0.6651
0.6710
0.7027
0.7406
0.8894
1.0968
1.3091
1.5149
1.6927
1.8/7l
1.9114
2.0280
2.1453
2.2850
2.4071
2.5383
2.6166
2.7423
2.8719
NAV•
SPECTRAL
VALUES
PERIOD
0.1154E
O.1020E
0.9173E
04
04
03
0.6800E
0.6700E
0.6600E
0.7657E
03
O.640QE
0.7064E
0.654•E
0.6075E
0.5651E
0.5263E
0.4903E
0.4565•
0.4247E
0.3944E
03
03
03
03
03
03
03
03
03
0.3652E
0o3368E
0.3090E
O.Z814E
0.2537E
0.2257E
0.1969E
0.1671E
0.1517E
0.1198E
0.8763E
0.7270E
03
03
03
03
03
03
03
0.6300E
0.6200E
0.6100E
0.6000E
0.5900E
0.5800E
0.5700E
0.56001
0.5500E
0.5400E
0.5300E
0.5200E
O.blOOE
0.5000E
0.4900E
0.4800E
0.4700E
0.4650E
0.4550E
0.4450E
0.4400E
O,6000E
0.5025E
0.4317E
0.3795E
FOR
MODE
C
03
03
03
02
02
02
02
02
02
LO
0ø00500
DEP[H
11.00
29.50
43.50
62.50
100.00
16g.50
225.00
300.00
400.00
500.00
600.00
700.00
800.00
900.00
1025.00
1200.00
1400.00
5
6
8
9
10
II
13
14
15
16
1600.00
18
1800.00
2000.00
2200.00
19
20
2l
2400.00
22
23
2600.00
LOVE HAVE SPECTRAL VALUES FOR MODE L1
u
0.4350E
0.4300E
0.4250E
0,4200E
1
2
PERIOD
Ol
0.6137E
01
01
Ot
01
Ol
OL
o1
01
01
01
01
0.5922E
0.5727E
0.5388E
0.5240E
0.5106E
0.4981E
0.4867E
0.4762E
0.4666E
0.4579E
01
01
01
01
01
01
01
01
01
01
01
0.4500E
Ol
01
01
0.4431E
0.4371E
01
01
01
01
01
Ol
0.4321E
0.4280E
0.4250E
0.4228E
01
01
01
o1
01
0.4215E
01
01
0.4209E
01
01
01
01
Ol
01
0.4209E
0.4210E
0.4208E
0.4186E
0•4155E
01
01
01
01
Ol
01.
0.4099E
01
Ol
01
01
0.4017E
0.3919E
0.3819E
o1
01
01
C
O.1700E 03
0.1•87E
03
Ool3IOE 03
0.t157E
0.10176
o.ee53E
0.7562E
0.5232E
0.4964E
0.4•3•'E
Ol
01
OL
03
•.SeOnE Ot
03
02
02
n.56OCE el
r.5460E 01
',.5200E 01
0.•550E
0.44116
0.431tE
0o4244E
Ot
01
01
0,6255•. 02 ..O.•00E
0.48746
0.4120E
0.•795E
0.345CE
0.3073E
U
0.64•0E
01
0.6200E nl
Oo6COOE01
02
02
02
02
02
Ot
n.4800E Ol
0.47CPE 01
•.•660E
01
0.4210E
0.4231E
•.•246E
0.•620E
0.4265E
31
O•
01
Ol
0.458CE
0.2410E 02 t•.4520E Ol
0.1847E OZ P.4480E 0l
0.4336E
•.4_367•
0.1502E
0.1408E
0.1311E
0.1215E
0.1127E
0.1062E
0.4383E
0.•38•E
0.4381E
0.4370E
0.433lE
02,
02
•2
02
02
02
Ot
0.4455E
fl.•450E
0.4445•
P.4440E
0.4435E
01
01
01
01
O!
01
01
01
nl
01
Period, sec
500
500
•
•50
•5
•
•
•oo
80
66.7
{
ø•' • •' I,,,,
h--•31
I0
I , l,, ,11• ,
,foo
"•"E
I00
:• •o
50
5.28
50
•oo
i0•
4
6
8
I0
12
14
Frequency, mc/sec
Fig. 2. Spectrumof Love wavesinglettransfer
functionfor severalvaluesof the sourcedepth.
500
0
I00
200
300
400
500
600
ILlN0, cm3•Zsec
(YIere and in the followingfiguresreferenceis
Fig. 3. Variation of Love wave singlettransfer
made to the fundamentalmode unlessotherwise function with depth of sourcefor various pestated.) L -- 10•'• dyne-sec.
riods.
RADIATION
PATTERNS OF SEISMIC SURFACE WAVES
Period,sec
500
200
2611
0
I00
667
..•
G(h)
-0.6
_
400
-05
-04
•
zoo
-03
$$
5•
-02
-40
-0.1
5
I0
Frequency, mc/sec
15
0
40
80
ILlNrr,cm•'zsec
120
160
200
20
Fig. 7. Variationof Rayleighwavefirstsinglet
Fig. 4. Spectrumof Love wavedipolaftransfer transferfunctionwith depth of sourcefor various
function for severalvalues of the sourcedepth.
+04
0
+0.2
0
periods.
-0.1
Period,
500
250
125
sec
I00
80
66.?'
io
20
•
h=lG5
km
264
50
j lOO
396
50
8
50o
-0.7
-0.6
-0.5
-0.4 6 -0.3
-0.2
-o.i
o
102
4
6
8
Frequency,
I0
12
14
mc/sec
Fig. 5. Variationof Lovewavedipolartransfer
of Rayleighwavesecond
sinfunctionwith depthof sourcefor variousperiods. Fig. 8. Spectrum
The corresponding
variationof the shearwave ve- glet transferfunctionfor severalvaluesof the
source depth.
locity with depth is also shown.
Period,
500
250
sec
125
I
I00
80
66.7
I
O_ 400___
I0 _
2O
I00
i,o
.•1oo
po
•-5
-I0
5oo
-50
2
4
6
8
Frequency,
I0
12
14
o
IOO
mc/sec
Fig. 6. Spectrum
of Rayleighwavefirstsinglet
transfer function for several values of the source
depth.
200
5oo
ILlNrz,crn3/•sec
4:00
500
Fig. 9. Variationof Rayleighwavesecond
singlet transferfunctionwith depth of sourcefor
various periods.
2612
5OO
BEN-MENAItEM
Period,
200
AND
HARKRIDER
sec
0
I00
66.7
50
I0
2O
A(h)
h =!0 km
+0.6
o•50
•1oo
+0.3
!00
5oo
I
-kO
200
0
I
I
B(h)
+l.O
-
300
Fig. 13. Variation of the Rayleigh wave first
dipolar function with depth of sourcefor various
5
I0
15
Frequency, mc/sec
2O
periods.
Fig. 10. Spectrum of Rayleigh wave ellipticity
function for several depths.
Period,
500
200
I00
sec
66.7
50
1.4
C(h)
io
_
5O
5
f•"-I
-0 6
't'
I
-o. 4
I,"
-0 2.
I
f
I
0
A(h)
+0 4
+0 6
I
I0
15
Frequency, mc/sec
-
+0.8
Fig. 11. Variation of Rayleigh wave ellipticity
function with depth for various periods.
20
Fig. 14. Spectrumof Rayleigh wave seconddipolar function for several values of the source
depth.
Period, sec
5OO
2OO
I00
66.7
B(h)
+1
1oo
-2
l
5
I0
Frequency,
J_qJ
15
mc/sec
Fig. 12. Spectrumof Rayleigh wave first dipolar
function for severalvalues of the sourcedepth.
L4
Lo
C(h)
o.5
o
Fig. 15. Variation of Rayleigh wave seconddipolar function for various periods.
RADIATION PATTERNS OF SEISMIC SURFACEWAVES
Period,
500
200
Period,
sec
I00
200
66.7
I00
2613
sec
66.7
50
I(
•1•
km
0.4•ø('
627'
km
$oo
0.2
o •1
I •'"'-,•'•1, I I I
0.5
0
I
5
I0
15
20
Frequency, mc/sec
5
I0
15
Frequency, mc/sec
Fig. 19. Spectrumof (Love first-mode/Love
Fig. 16. Spectrumof singlet(Rayleigh/Love)excitation ratio for several values of the depth.
fundamental-mode) excitation ratio for several
values of the depth.
,') .....
200
..........
KRNrr
0.4
0.2
0
08
06
KE Nrz
Fig. 17. Variation of singlet (Rayleigh/Love)
o
excitation ratio with depth of sourcefor various
Fig. 20. Variationsof (Love first-mode/Love
periods.
fundamental-mode) excitation ratio for various
periods.
o
io
20
•
The relative excitation functionsof Rayleigh
and Love wavesare shownin Figure 16. The top
figurerepresents
the true spectraldisplacement
50
ratio recorded at azimuth • = 45ø due to a
horizontalforce at depth. The variation with
•'100
depthof No(•)(h,•), the Lovewavefirst-mode
singlettransferfunction,
is shown
in Figure18.
5O
The resemblance
to Nrr(h, •) (Figure 7) is
apparent.
The relative excitationof the two Love modes
5O0
I
-200
I
is givenin Figures19 and20. For a singleforce
-I00
(i)cm•2seOc
(ist
' Mode)
+100 +200at depth,thisratio will dependneitheron the
ILlNe
orientation of the force nor on the azimuth of
Fig. 18. Variation of the Love wavefirst-mode
singlettransferfunctionwith depth of sourcefor the stationand will representthe true spectral
displacement
ratio.
various periods.
2614
BEN-MENAHEM
AND HARKRIDER
The functionsN0, N•,, and N,r have the
Auxiliary parametersand nodallines. A simdimensions
of k•/'1•-•. Sincek• was expressed
in plificationof the expressions
for the displacement
km-• and/• = p/3'.in (g/cm3)(km/sec)
'•, a con- field is achievedupon the introductionof a
versionfactor was appliedto convertdisplace- parameter •/defined as
mentsto cgsunits (Figures2, 3, and 6 to 9).
cosy = cosX(1-- sin2Xsin2 $)-•/•'
GENERAL PROPERTIESOF THE I•ADIATION
The geometrical
interpretationof V is clearfrom
SymmetryreMtions. It is useful to look for
symmetry with respect to the variable • and
the parametersX and • in order to minimize the
Figure1. We notealsothat • = V -- •/2 is the
angle between the two dip directionsof the
planeof motionand the auxiliaryplane.The dip
angle of the auxiliary plane, $', is given by
cos$' = sin k sin $. From this relation,together
with the definitiongivenabove,we find that
tabulation of the azimuthal distribution of the
modulusIx(•)I and the spatialphasearg x(•).
We denote by m•(•)e•
and m•(•)e•
the
respectivepolar representationsof the radiation
patterns for Rayleigh and Love waves. It then
sinv = cot • cot•' = cos•
followsdirectlyfrom (94), (•5), and Table 1 that
•(s + •;•,
•,•, •) = •(s;•,
(46)
sin V = sin• cos•(1 -- sin"• s• • •)-'/•
PATTERNS
•,•, •)
cosX= •cosp•cosy
(•0)
s•p
for all coupleand double-couple
displacements.
Equation 40 can alsobe written as
(47)
= sinXsin $
With the aid of these angleswe can transform
(7) and (8) into
mR.L(O
q-•v)= m•.L(0) (41) Uo• =
•R[No(h)e
•s•/• [cosp[ s• (O--V) (48)
•(o + •) + •(o) = •(o + •) + •(o) = •/•
The relationsgivenin (41) are due to Aki [1964]
and are useful for checking the calculations.
ß[•(a) [•o• •[ •o• (0 - v) + i •in •]
It
A nodalline for Love wavesexistsa• 0 = •,
and i•s angular positionis independen•of •he
constantsof •he medium. The spatial phaseis
either 3•/4 or 7•/4. For Rayleighwavesa
nodallinemay exis•onlyif N•(h) = 0 or p = 0.
The phasewill generallydependon •he s•c•ure.
Likewisewe obtainfor a couplesource
is assumed here that the source time de-
pendenceis a Dirac delta function. The antisymmetry betweenright-reverseand left-normal
faultings yields
U(X + •) = --U(X)
(4•)
Furthermore,
U0c = nI [R[kLNo(h)e
-'•'/4 [cos
Plsin(0 -- V)[--sin$sin0 + iG(h)cos$]
(4.
(60)
U,c ----« Inl[R[kRNrz(h)e-ia•'/4[[cos
p[sin(•{A(h)sin(20-- •])-[-B(h)sin•}
+ 2iX(h) sin (0 + •')]
U(--$) = --U($)
U,(7r-- •, •r -- 0) = U,(k, 0)
(43)
(51)
where
• •]•'
(44) X • = sin• },[1 -- C(h)cos
vo(• - x, • - o) = - vo(x, o) (45)
which can be established from the nature of the
coefficients
in Table 1. The foregoingrelations
show that it is sufficient to know the radiation
q- cos•'
k cos
• $(1 -- C)2
sin•' ----[(1 -- C)/[X[] cos)•cos$
(52)
(53)
cosf = [(1 -- C cos
•' $)/X]sinX
pattern for 0 _• 0 •_ •r, 0 _• /l _• •r/2, and andthe valueof •' is determined
by the algebraic
0 _• X _• •r/2. Thevaluesforthesupplementarysigns of the two expressions.Thus for Love
anglescan be obtainedfrom theseby reflections. waves we always find at least one nodal line.
RADIATION PATTERNS OF SEISMIC SURFACE WAVES
Fig. 21. Radiationpattern from two dipolar
sourcemodelsrepresentinga left-reversefault at
zero depth for a spectralperiod of 106 see (numerical amplitudescalein all figuresis for the
reduced displacements).
2615
Fig. 23. Radiation pattern from a doublecouplesourcemodelrepresentinga normal fault,
at two distinct depths.Note changein Love wave
radiationpatternsas sourceis displacedfrom the
free surface to base of crust.
Fig. 24. Radiationpattern of Rayleighwaves
from two double-couplesourcemodels represent-
ing left-reverse
faults,eachat twodistinctdepths.
Fig. 22. Radiationpatternfrom two dipolar Note strikingvariationsof radiationpattern in
sourcemodelsrepresentinga left reversefault
at zero depth for a spectralperiod of 167 sec.
both casesas positionof sourcechanges
from free
surface to base of crust.
2616
BEN-MENAItEM
AND
The existence
of the secondnodewill dependon
the dipangle.For highermodesG(h)mayvanish,
and a clover-leafpattern will be producedfor
someparticularcombinations
of frequencyand
sourcedepth, even for dip angleswhich differ
from•/2.
The Rayleigh wave pattern is more involved. There will be no nodes unless • = 0 or
k = • = •/2. The shapeof the patternwill
dependmainly on the relative magnitudesof
.4, B, and C. One may expect'almostnodes'for
frequency-depth combinations which render
HARKRIDER
Unlike the previouscasethere will, in general,
be no nodal lines for either Rayleigh or Love
pattern, and the existenceof 'almost nodes' will
dependon the relative magnitudeof the functions of the medium.The case• = 0 will give a
singlenode,and the casek = •/2, B = 0 will
give a quadrant pattern for both waves. A
surfacesourcewill give a quadrant pattern for
Love waves independentof the structure and
the frequency.For Rayleighwavesthere will be a
nodeforh = 0 onlyif Isin71--• (1 -- •0)/(1 •- •0).
Expressing this condition in terms of the dip
and slip angleswe find that the dividing line for
C >> A,B or A • B,C. The caseB • A,C
will give an almost circular pattern. For the
•0 - 1/4 is represented
in the (k, 8) planeby
double-couplesourcewe find
the curve
Uo
dc-- InlIRIkLN•(h)e-'S'/'[]
cosplsin• cos(2o- •) + ia(h)Ico qlsin(0 N
'h)
ud• = InlIRI• ,,• •e
[Icos
p[sin$[A(h)s• (20 -- .) + B(h)s• •]
- ice)Icos ql cos(0-
(•4)
•)1
(•)
where
sinX = 3(25 -- 16sin"/•)-•/"
cos• cosX = cosT [cos
cosT = cosk cos$(cos"k cos
•' $
q- sin•'X cos"2•) -•/"
(•)
(57)
Dependence
oj' patternson the sourceelements.
The modulusIX( 0)l describes
in the generalease
an obliquely symmetric, dosed curve of the
sixth order, otherwiseknown as the hypertroehoid.Its shapeis governedby the coefficients
d• (Table 1) which in turn are compositefune-
Fig. 25. Top: Radiation pattern of Rayleigh
Fig. 26. The effectof sourcedepthon the Raywaves,coupleversusdouble couple,for a reverse leigh wave radiation pattern from a left-reverse
fault at depth of 66 km. Bottom: Radiation pat- fault with a double-coupleforce system. Note
tern of Rayleigh waves, couple versus double radical changeof radiation pattern as sourceis
couple, for a horizontal fault.
displaced from the free surface to base of crust.
RADIATION PATTERNS OF SEISMIC SURFACE WAVES
2617
phase
in a unitring[Brune,1961;Haskell,1963].
The amplitudeplotted in thesefiguresis the
reduced
amplitudeIU*I, whichis a normalized
entityof dimensions
cm•l•'/dyne.
It is relatedto
the true spectralamplitudeIUI throughthe
equation
-•9R(•0)l
inI U*
Ivl=10
I Ie'
(micron-see)
(58)
IR(•)I is the magnitudeof the forcein cgs
units, r is the epicentraldistancein km, and
y(•) is the absorptioncoefficient
in km-•. The
analytic expressions
of the patternsshownin
Figures21 to 29 aregivenin Table3. In choosing
the variousexamplesit wasintendedto emphasizestrongvariationsof patternswith respectto
depth of source,frequency,mode, and force
Fig. 27. Top: The effect of sourcedepth on configuration.
Note the stationarybehaviorof
radiation pattern of Love waves from a left- the phasesat azimuthswherethe amplitudeis
reversefault (doublecouple).Bottom: The effect
of sourcedepthon radiationpatternof Rayleigh maximum and its rapid variation at azimuths
wherethe amplitudeis relativelyminimum.This
waves from a left-reverse fault (single couple).
developsinto an extremecaseat whichaiump
tions of the source constants(X, $, h), the
of w radians occurs at the nodal lines. The
periodT, and the physicalconstants
of the phasesin all figuresvary betweenzeroand 2•r
earthmodel.To studythe effectsof the variation radians. The additional constantphasefor each
of theseparameters
on the radiationpatternwe caseis given in Table 3.
have plottedsometypicalexamples.
(Figures
21 to 29). Azimuthalvariationof amplitudeis
DISCUSSION AND CONCLUSIONS
Previous endeavors in this field were restricted
at the centerof the figure.The peripheralcurve
showsazimuthal variation of the corresponding either to simplertypesof mediumsor to sources
J,,'h
\ \ 0Iol /
J,k• x \
l •
m
j
/
/
-'-DC
• '• I I / / ••
Ur
, x \ ßT=125sec
/
ø••Oj
o)j• j JJ
• I •
/
180
k
/ I •• •
•-90
•.
I
•
/
, x
/ , • • •
_ x x i I /_/
x-x •
J'
JJJ•J JJJJ•
•/•
t
I
• ,
,
• I • • ß
,,,
Fig.28. Theeffect
ofthesource
frequency
ontheradiation
pattern
ofRayleigh
waves
from
a reversefault (doublecouple)at depthof 100kin.
2618
BEN-MENAHEM
AND
HARKRIDER
frequency,
dip and slip angles,andfinallythe
constants of the elastic medium.
Radiationpatternswerecalculated
forperiods
lessthan350sec,sincewebelieved
thata sphericity correction
for longerperiodswasimperative.
It would be advantageousto recalculatethe
transfer functions of the medium for a multilayered spherical earth whenever a suitable
computer program becomes available. It would
alsoproveusefulto studythe sensitivity
of the
transfer functionsto variations of the constants
of themedium.
The caseof an oceanic
pathis of
particular interest.
The method given here can be extended to
encompass
a widerclassof sources.
Multipolar
sourcesmay be obtainedby applyinga differentialoperator
of a suitable
degree
to thesinglet
field. A horizontal extension of the source
Fig. 29. Dependence of radiation pattern of
Love waves upon the displacementmode for two
introduces
a frequency-dependent
asymmetry
factorinto the displacement
spectrums,
and a
verticalextension
will necessitate
an integration
of thedisplacements
withdepth[Ben-Menahem,
different sources.Top: Vertical fault (double couple). Bottom: Strike-slip fault (double couple).
of lesser complexity, or both. In the present
study we have made an effort to overcomethese
unnecessary
limitations. As usual, this requires
increasing the number of the participating
physicalparameters,suchasthe depthof source,
Period, sec
555
200
145
1961]. Other sourcerepresentations,
such as
proposed
by Knopoffand Gilbert[1960]and Aki
[1964],canalsobe treatedby our method.
The radiationpatternis an importantsignature of the seismicsource.It can and shouldbe
studiedmoreintensively,therefore,in orderto
III
Period, sec
553
200
143
III
..
8:,•0•
i00,)
/
4
6
Frequency, mc/sec
8
2
_
4
6
8
Frequency, mc/sec
Fig. 30. Variationof thereduced
amplitude
spectrum
withazimuth(fromstrikeof fault) for
samesource
asin Figure28. Note progression
of spectral
nodetowardshorterperiodsasazi-
muth changesfrom 0 ø to 90ø.
RADIATION
TABLE
3.
PATTERNS
OF SEISMIC
SURFACE
2619
WAVES
Radiation Pattern Coefficients of the Reduced Displacements
U* -- Uoe-•o[ao• bosin (20 - 00) • ibl sin (0 • 01)]Shownin Figures21 to 29
The angle•0 hasbeensuppressed
in the phase-ring
diagramsof thesefigures.
U0,
Figure Wave* Source• cml•2/dyne ao
00,
0•,
•0,
bo
bl
deg
deg
rad
(1)
(2)
(3)
(4)
(1)
(2)
(3)
(4)
R
R
L
L
R
R
L
L
C
DC
DC
C
C
DC
C
DC
148.0
296.0
418.0
214.0
22.7
45.4
35.6
71.3
.46
.46
0
.96
1.57
1.57
.27
0
i
i
I
-1
1
i
-1
I
2.08
0
0
0
10.20
0
0
0
16.4
16.4
-73.6
-73.6
74.5
74.5
-15.5
-15.5
31.1 3•r/4
0
3•r/4
0
3•r/4
0
-•r/4
14.5 3•r/4
0
3•r/4
0
-•/4
0
3•r/4
23
(1)
(2)
(3)
(4)
R
L
R
L
DC
DC
DC
DC
270
447
74.0
375.0
1.6
0
.15
0
-1
1
i
I
0
0
10.00
1.24
-90
0
-90
0
0
0
0
90
24
(1)
(2)
(3)
(4)
R
R
R
R
DC
DC
DC
DC
620.0
537.0
173.1
150.0
1.15
1.63
.10
.14
i
-1
-1
i
0
0
-2.04
0
45
-90
45
-90
25
(1)
(2)
(3)
(4)
R
R
R
L
C
DC
DC
DC
71.2
142.4
866.0
563.0
I
1
0
0
0
0
-7.32
0
i
I
0
0
0
0
26
(1)
(2)
(3)
(4)
R
R
R
R
DC
DC
DC
DC
831.2
232.0
162.1
85.5
.46
.04
1.04
.91
I
- 1
i
I
0
1.89
2.53
1.86
16.4
16.4
16.4
16.4
27
(1)
(2)
(3)
(4)
L
L
R
R
DC
DC
C
C
1161.0
375.3
116.0
81.0
0
0
.04
1.04
i
I
-1
i
--.23
--.22
5.79
4.89
-73.6
-73.6
16.4
16.4
28
(1)
(2)
(3)
(4)
(5)
(6)
R
R
R
R
R
R
DC
DC
DC
DC
DC
DO
104
3.68
i
I
0
0.62
i
i
0
i
i
--1
--1
0
0
0
0
0
0
90
0
--90
--90
--90
--90
0
0
0
0
0
0
--•r/4
--•r/4
--•r/4
--•r/4
3•r/4
3•r/4
(1)
(2)
(3)
(4)
L
L (•)
L
L (•)
DC
DC
DC
DC
538.0
142.8
538.0
142.8
-- .25
i
-- .25
i
--90
0
--90
0
90
90
0
0
3•r/4
--•r/4
3•r/4
--•r/4
21
22
29
6
71 2
28 0
160
57 5
11 5
0
0
0
0
0
0
I
0
i
0
0
0
-45
0
-•/4
-•r/4
3•r/4
-•r/4
3•/4
3•r/4
-•r/4
-•/4
0
0
0
90
-•/4
-•r/4
-•r/4
3•r/4
0
136.8
136.8
136.8
3•r/4
-•r/4
-•/4
-•r/4
46.8 3•r/4
46.8 3•/4
12.7 -•/4
1.2 -•/4
* L - Love, R - Rayleigh.
• C - couple,DC • doublecouple.
We wish to thank Keiiti Aki for valuable diseliminatethe guessworkinherent in the firstcussionsduring the early stagesof this work.
motionmethodand to put seismicsourcestudies
REFERENCES
on a firm physicalbasis.
Acknowledgments.This researchwassupported Aki, K., Study of Love and Rayleigh waves from
earthquakes with fault plane solutions or with
by contract AF-49(638)-1337of the Air Force
Office of Scientific Research as part of the Advanced Research Projects Agency project Vela.
known faultings, Bull. Seismol. Soc. Am., in
press, 1964.
2620
BEN-MENAHEM
AND
HARKRIDER
Ben-Menahem, A., Radiation of seismic surface
waves from finite moving sources,Bull. $eismol.
$oc. Am., 51, 401-435, 1961.
elastic half-space, in preparation, 1964b.
Haskell, N. A., The dispersionof surfacewaves in
multilayered media, Bull. $eismol. $oc. Am., 43,
Ben-Menahem, A., and M. Nafi ToksSz, Source
17-34, 1953.
mechanism from spectra of long-period seismic Haskell, N. A., Radiation pattern of Rayleigh
surfacewaves, 2, The Kamchatka earthquake of
waves from a fault of arbitrary dip and direction of motion in a homogeneousmedium, Bull.
November 4, 1952, J. Geophys. Res., 68, 5207$eismol. Soc. Am., 53, 619-642, 1963.
5222, 1963.
Brune, J. N., Radiation pattern of Rayleigh waves Knopoff, L., and F. Gilbert, First motions from
from the southeast Alaska earthquake of July
seismicsources,Bull. $eismol. $oc. Am., 50, 11710, 1958,Publ. Dominion Obs., Ottawa, 24(10),
134, 1960.
Yanovskaya, T. B., On the determination of the
1-11, 1961.
dynamic parameters of the focus hypocenter of
Harkrider, D. G., Surfacewavesin multilayered
an earthquake from records of surface waves,
elasticmedia, 1. Rayleigh and Love wavesfrom
Izv. Alcad. Naulc S$$R, $er. Geo/•z., 209-301,
buried sources in a multilayered elastic half-
space,Bull. $eismol.$oc. Am., in press,1964a.
2. Rayleigh and Love wave seismograms
and
spectrafrom buried sourcesin a multilayered
1958.
(Manuscript receivedFebruary 21, 1964.)