JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 88, NO. B12, PAGES 10,285-10,298,DECEMBER 10, 1983
Observationsof Coupled Spheroidaland Toroidal Modes
GuY MASTERS,JEFFREYPARK AND FREEMAN GILBERT
Instituteof Geophysics
andPlanetaryPhysics,
Scripps
Institutionof Oceanography,
University
of California,SanDiego
In low-frequencyseismicspectrathe Coriolisforce theoreticallycausesquasi-degenerate
coupling
of spheroidaland toroidal modeswhosesphericalharmonicdegreesdiffer by unity and whosefrequenciesare close. It has been found that the couplingcausesclearlyobservableeffectson seismic
spectrabelow 3.0 mHz. Many fundamentaltoroidal modeshave a spheroidalcomponentand vice
versa.For example,
the 0St•modesfor If= 8-22 aresignificantly
coupled
to the 0Tt•modesfor
If = 9-23. The mean frequenciesof pairsof coupledmultipletsare repelled,and the mean attenuations are averaged. The shifting of complex frequenciesmust be adequatelyencompassedin the
constructionof sphericallyaveragedand asphericalmodels of the density and elasticand anelastic
structureof the earth. The couplingalso causesvariationsin multiplet amplitudeswhich result in
spectralpeaksat toroidal mode frequencieson vertical and radial componentsand at spheroidal
mode frequencieson transversecomponents. The variations must be taken into account in constructingGreen's functionsfor the study of low-frequencysourcemechanisms.
1.
INTRODUCTION
Splitting in low-frequency spectrawas first observed in
the analysesof the recordingsof the great Chilean earth-
will be limited by our ability to model the couplingeffects.
Techniques for handling the computational problems
associated with mode coupling are discussed in the next
section. A more complete discussion will be given else-
quakeof May 22, 1960 [Nesset al., 1961; Benioffet al., where (J. Park et al., manuscriptin preparation,1983). A
1961]. It was readilyapparentthat the observedsplitting comparison is made between computed coupling effects
was causedprimarily by the rotation of the earth [Backus and observedeffectsin IDA (InternationalDeploymentof
and Gilbert, 1961; MacDonald and Ness, 1961; Pekeris et Accelerometers)data and GDSN (Global Digital Seismic
al., 1961]. Coupling among multipletswas first treated Network) data. The computationis done via a Galerkin
theoreticallyby Dahlen [1969], and the effectsof attenua- matrix method and includes rotation, ellipticity of figure,
tion in couplingwere shownby Woodhouse
[1980]. As the and attenuation. In some cases, nonaxisymmetric asphericalculations and observations reported here show, the
importance of coupling is much more pervasive then previously realized.
Measurements of frequencies and attenuation rates of
free oscillation multiplets are usually interpreted in terms
of a spherically averaged earth structure. This procedure
cal structure is included. The overall agreement between
the computations and the observations is generally good
but reveals the desirability of better spherically averaged
and asphericalearth structuresfor modeling low-frequency
seismograms.
2.
can be justified if the multiplets are isolated and departures of the real earth from the sphericallyaveraged earth
are small [Gilbert,1971]. In this paperwe showthat for a
more realistic model of' the earth, one that is rotating and
has small aspherical perturbations in structure, the
assumption that a multiplet is isolated is often wrong and
significantcoupling of multiplets occursover a large part
of the spectrum. The consequencesof the coupling are
RESUM1}OF
THE COMPUTATIONAL
PROCEDURE
The Galerkin procedure that we have applied is closely
related to the variational method for mode coupling
describedby Luh [1973]. In fact, in the absenceof
attenuation the equations are self-adjoint, and the Galerkin problem exactly coincides with the variational problem.
Luh's formulation, in turn, derives from the correspontwofold: (1) elastic and anelasticsphericallyaveraged dence between variational methods and first-order perturmodels of the earth are biasedunlessthe couplingeffects bation theory [Moiseiwitsch,i966]. First-order perturbaare included, and (2) source mechanismretrieval from tion theory based on Rayleigh's principle has been
low-frequency data is biased unless coupling effects are developedby Dahlen [1968, 1969], Madariaga [1971],
included in the computation of the Green's functions.
Zharkov and Lubimov[1970a,b], Woodhouse
[1976], and
The conclusionsof this paper would be merely academic others. The relevant formulae are refined and summarif the effects of coupling were not obvious in lowizedby Woodhouse
andDahlen[1978].
First-order perturbation theory has been used to calcufrequency data. The dominant signals in low-frequency
seismogramsare the fundamental spheroidaland toroidal late the splitting of isolated multiplets and the coupling
mode multiplets. These multiplets are coupled together between singlets of' degenerate multiplets. Variational
over a large frequency band, mainly by the Coriolis force,
methods have been proposed to describe coupling without
leading to very obvious effects in the data. It is clear, attenuation over many different multiplets [Luh, 1973;
from the examples we give, that our ability to resolve
Morris and Geller, 1982; Kawakatsuand Geller, 1981].
earth structure and source mechanismsat low frequencies However,calculations
reportedby Woodhouse
[1980] indicate that attenuation significantlyalters coupling between
Copyright 1983 by the American GeophysicalUnion.
stronglycoupledmultipletswith different intrinsicattenuation rates. To include attenuation, the self-adjoint variaPaper number3B1411.
0148-0227/83/003 B- 1411$05.00
tional methods must yield to more general Galerkin pro10,285
10,286
MASTERSET AL.:OBSERVATIONS
OF COUPLEDMODES
cedures. The bonus of a more general procedure is that
we can predict perturbationsin apparent multiplet attenuation rates for comparison with measurements of average
attenuation.
In the Galerkin procedure [Kantorovichand Krylov,
1958; Mikhlin, 1964; Finlayson, 1972; Marchuk, 1975;
Baker, 1977] for an integrodifferential,
linearoperator,H,
one chooses
a sequence
of basisfunctions{•b•,•b2, ... }
that satisfy the appropriate boundary conditions, possess
the degree of smoothnessrequired by H and that can be
extended to a complete set. Any finite subset of the
sequencemust be linearly independent.
A solution to the eigenvalue problem
H•b = h•b
(1)
can be approximated within the subspace spanned by
{4•, 4•2, ''',
any two eigenfunctions
si, s/of H0
(si',H• s/) = V0 - toW0 - to•T0
(7)
where to representsthe eigenfrequencyto be determined.
In (7) we have suppressed
any effectsof physicaldispersion [Akopyanet al., 1975, 1976; Liu et al., 1976;
Kanamoriand Anderson,1977]. Includingphysicaldispersion is important for interactions over a large frequency
band, but for most interactions over small frequency
bands, the center of the band can be taken as the reference frequency and physical dispersion can safely be
neglected(J. Park et al., manuscriptin preparation,1983).
The T0 termsin (7) are properlyplacedin the kinetic
energymatrix T in (2), whichis Hermitianbut not necessarilydiagonal.The Coriolisforceleadsto the matrix
in (7) and formally requiresthe solutionof the quadratic
4•.} by calculatingthe interactionamong eigenvalue problem
the basisvectorsthroughthe operatorH. In this way (1)
(V - tow - to2T)x = 0
is converted into the n x n algebraiceigenvalue problem
Vx = hTx
(2)
(8)
By defininga vector y = tox, the quadraticproblem (8)
can be converted to a linear algebraic eigenvalue problem
where
E/= (4•[,H4•/), T0-- (4•[ 4•/)
(3)
and ( , ) denotesan inner product,usuallyan integration over the volumein whichH is defined,with •b' being
the complexconjugateof •b.
We take as our basis set the elastic-gravitationaleigenfunctions of a spherically symmetric earth model, usually
[Gatbowet al., 1977 pp. 49-50]
T
O
O
I
(9)
at the expense of doubling its size. At frequenciesbelow
about 0.5 mHz, modes with relatively very different fre-
quenciescouplestrongly,and the form (9) must be used.
the monopoleof H. (At very low frequencieswe include
Such strong couplingis found to exist for •S•, 0S• and the
in our basis set the inner core toroidal modes, the outer
core toroidal and gravitational elastic modes, and the secu-
Chandler wobble (primarily 0T1 in shape) [Smith and
Dahlen, 1981], for example. However, in the present
lar modes, (iii)-(vi), describedby Dahlen and Sailor paper the observations of interest lie above 1 mHz, and
[1979,p. 615]. This is doneto ensurecompleteness.)
The the dominant couplingis confinedto frequencydifferences
eigenfunctionsare vector fields defined within the earth
about the same order of magnitude as the rotation rate
volume V and the associatedall-spacegravitational poten-
(11.606/xHz). Thus we can apply (8) to narrow fretials. The vector fields are denoted,Sfm,where n is the quencybands. We have found that we canevaluate(7) in
radial overtone number, f is the degree (total angular narrow frequency bands by approximating the term linear
order) of the spherical harmonic, m is the azimuthal in to
order,and q is (S, T) for a (spheroidal,
toroidal)mode.
The operator H can be written as
H-
H0+ H1
(4)
where H0 is the sphericalaverageof H (the monopole)
and H• representsrotation, ellipticity of figure, attenuation, and asphericalstructure. The basisset is normalized
(.•qm
'..S•'qm'.)
=f•dV
Oo(r)
.•qm
'' .S•'qm"
-- an aft'•mm'aqq'
(5)
wherepo(r) is the densityfor H0 asa functionof radiusr.
With the chosen normalization
(n•qm
*, So nS•'qm
") -- (n•q)2an a•' amm'
aqq'
(s,',Hls•) = Vo - •; - to•To
(1O)
Uo = (to,to/)'/'
Wo
(11)
where
This approximation does not significantly effect the
numerical results for frequencies above 0.5 mHz.
With the indicated approximations the Galerkin procedure in narrow frequency bands is reduced to the general algebraiceigenvalue problem
(V - toeT)x - 0
(12)
where T is Hermitian positive definite and V, becauseof
attenuation, is complex but not Hermitian. The system
(12) is solved numericallywith slight modificationsof
(6)
EISPACK routines (Smith et al. [1976] and see Garbow et
where ,gq is the degenerateeigenfrequency
of the multiplet (n,g, q). When [IHi[[ << [IY0[I,V in (2) and (3) will
al. [1977, pp. 49-50] for higher degreeeigenvalueproblems,e.g., (8)-(9)).
be diagonallydominant.
We have used the formulae of Woodhouse[1980] to calculate the interaction due to rotation, ellipticity of figure,
attenuation, and, in some cases, asphericalstructure. For
In this paper we are primarily concerned with coupling
among fundamental spheroidal,0S•, and toroidal, 0T•,
modes. Coupling among fundamentals and overtones is
small and has been neglected in the frequency band
MASTERS
ETAL.:OBSERVATIONS
OFCOUPLED
MODES
10,287
1-5 mHz. Someovertones,e.g. 7S• and 2T2, are strongly
In order to compareour couplingcalculationswith data,
coupled but are not discussedhere. Angular selection we calculatethe mean singletfrequencyof the hybrid mulrulesfor couplingby rotationand ellipticityare simple. A tiplets oSt•. The groupingof hybrid singletsinto distinct
mode nS• can coupledirectlyto n,T•+•, ,,S•, •,S•+2. A hybrid multipletsremainswell definedeven for profoundly
mode, T• cancoupledirectlyto ,,S•+•, n,T•,,,T•+2. Other, coupled modes like 0S•-0T•2. Only when the coupled
indirectcouplingsare possible[Smith,1977]. Rotationand multiplets are very nearly degeneratein both frequency
ellipticity do not couple different azimuthal degreesm. and attenuation does the distinct groupingdisappear. We
We have used this fact, when asphericalstructure is not determine the averageattenuationof the hybrid multiplet
included, to order the matrices V and T in (12) into block oSt• by averagingover the imaginarypart of the singlet
diagonalform to give a distinct eigenproblemfor each eigenfrequencies.
value of m. The eigenvectors
Xm for each m represent
We can gain insight into the qualitative features of
the hybrid singletsappropriateto the coupledproblem. In strong coupling with a simplified example. Consider coualmost all caseswe find that one modal type predominates plingbetweenzeroth-ordersingletsoS•m and 0T•. If we
in the amplitudeof the coupledsinglets. We use the nota- consider only attenuation and coupling due to Coriolis
tion (oSt•,oTs•) to representa coupledmultipletwhose force, the potentialenergycouplingmatrix takesthe form
majorcontribution
comesfrom a multipletof type (0S•,
0T•) and whoseminorcontribution
is from (oT•,,oS•,),
V -
where f' ;• f.
Our calculations
showthat Corioliscouplingis the dominant effect. Our approach,via an integralequation,can
(o,s)2
•
•
(COT)
2
(14)
be applied to a greater band of frequenciesthan can the
first-order perturbation approach [Luh, 1973, 1974].
where cos,cot are the (complex) spheroidaland toroidal
singlet eigenfrequenciesand • is the coupling between
oS•m and 0T•7• due to Coriolisforce. The hybrid eigenfre-
Furthermore, the vital effect of attenuation is included in
quenciesare defined by
our calculationsfollowing the formulation of Woodhouse
[19801.
To see why Coriolis couplingis dominant, we offer the
following hueristic argument. Consider the W matrix in
(7)-(9):
co2+
= cos2+coer
___
,/2((cos2_co2r)2
+4lel2),/2
(15)
2
SinceRe[(cose-coer)2
+ 41•121'/•
½ Re(cos2-coer),
the hybrid
eigenfrequencies
repel, as is well known from examplesin
mechanics [Sommerfeld, 1952, pp. 106-111].
Since
]Im[(o•s2-o•2r)e+4]ele]'/•l
•< ]Im(o•s2-o•)],the hybrid
W•2
=4f•po(r)
[sl.(illxs2)*
dV] (13)attenuations
will attract. One expectsthat strongcoupling
where s•, s2 are any two eigenfunctions
and 11 is the sidereal angular velocity vector. The vector cross product in
would tend to homogenize singlet attenuation rates. If we
assumethat Re(cos)> Re(for), define r = co+-cosand
(13) rotates(,S•, , T•) motioninto (, T•+•,nS•+•)motion /xro= cos-cot, we can express(15) as
quite efficiently. This leadsto strongcouplingin the 1.5
to 3.0-mHz frequency band where such modes are close in
frequency. In contrast,couplingthrough ellipticity can be
regardedas diffractionof sphericalwavesby the surfaceof
the earth.
The surficial wave number of most modes is
[r +
- 4cos
2 + •-•-
cos
We canexpresslel2 [Dahlen,1969,equation23]:
large comparedto that of the earth's equatorialbulge. In
(17)
fact, the ratio of wave numbersfor a mode of angular
4f2-1
order f is roughly f/2 (there is additionalvariation with
azimuthal order). We thereforeexpectthe diffractionof where D is the rotation frequency and a is a dimensiontoroidal motion into spheroidal motion and vice versa to
less scalar near unity. We can take cosfor co+in (17)
be small for ellipticityor any other very long wavelength without changingterms of highestorder and average(!7)
lateral structure, such as the transition zone model of
overall azimuthalordersIml Vto obtain
lel
2=4co+2f•ea
2f2--m2]
Masters et al. [1982].
Such diffraction is much less
efficientthan rotationas a couplingmechanism.
2 co•D2
2 f
(18)
This hueristicargument is in agreementwith the results
reportedby Luh [1974]. Usingquasi-degenerate
perturba{ /mdenotes
expectation
valueoverm. Wespecify
tion theory, he computed the effect of rotation, ellipticity where
of figure, and the continent-oceanasphericalstructureon /xcoto be constantover m. This is not generallytrue, but
Solving(16) for Irl andusing
multiplets very close in frequency. He found that Coriolis is a usefulapproximation.
result:
coupling between a spheroidal and a toroidal multipier (18) in placeof I•12givestheapproximate
overshadowsall other types of coupling. This means that
at the frequenciesconsideredin this report, aspherical
2 2 2 f t-(Ao•)
(19)
•-D
a (f+i/•--••--2
structure, including any of the tectonicallyregionalized
{E2}m
=T
a •-.l-'/2
structures[e.g., SilverandJordan,1981], is not as important as the Coriolisforce in couplingmultiplets. At higher If Re(cot)> Re(cos), the sign of r will be negative.
frequencies,above 3.0 mHz say, the effect of aspherical Equation(19) showshow mean frequencyshifts become
structure will become more important, and rotation can appreciable whenever the frequency difference between
eventually be neglected.
coupledmodes is of the same order as, or smaller than,
10,288
MASTERSET AL.:OBSERVATIONS
OF COUPLEDMODES
SpheroidoI-Toroidolmode frequencydifferences
150
features in Figure 1 vary little for many modern earth
models and imply strong Coriolis couplingfor •'= 10-20
and near •' = 32.
3.
100
COMPUTATION
AND OBSERVATION
OF FREQUENCY
SHIFTS
AND AMPLITUDE VARIATIONS
The calculationsof the precedingsection predict several
features that can be seen in low-frequency data. The
N
50
interactionis suchthat unlessthe coupled0Stand 0Tt+l
modes are nearly degeneratein their frequencies,singlets
are grouped together in hybrid multiplets which have
0 -
predominantlytoroidal(Ts) or spheroidal(St) characteristics. In the case of near-degeneracy,i.e., when the
singletsof couplingmultipletsare overlappedin complex
frequency, the separationinto hybrid multiplets is not
alwaysclear, and the coupledmultiplets (a super multiplet) must be considered
as a singleentity. Most earth
-50
models predict near-degeneracy for the multiplets
oSll-oT12, 0S19-0T20,and 0S32-0T31,as shown in Figure 1.
-100
We shall demonstrate that the first two are well observed
but that the latter is by no means obvious in the data.
-150
0
I
5
•
10
;
1
I
20
215 3 •0
35
The model we have chosenfor demonstratingthe coupling effects is model 1066A of Gilbert and Dziewonski
40
[1975]. This is an isotropicmodeland providesthe best
Sphericol hormonic deqree /.
Fig.1. Frequency
differences,
in tzHz,between
fundamental
0St, fit to the existing mode dataset for frequencies below
and0Tt'+lmodes
fort• = 7-28 andbetween
fundamental
0St,and about 5 mHz. The model is not optimal in that it was
0Tt'-I modesfor t• - 3-6 andt• = 25-39. The plussigndenotes constructedwithout considerationof the effectsof physical
valuesfor the anisotropic
PREM model[Dziewonski
andAnderson, dispersion and furthermore does not fit the fundamental
1981], and the dot for model 1066A [Gilbertand Dziewonski, spheroidalmode data well above angular order 30. A new
1975].
model is highly desirable but will have to await a more
complete evaluation of bias in the mode dataset.
the earthrotationfrequency(11.606tzHz). The frequency
To illustrate the coupling behavior, we show computed
differences
betweenneighboring
0S•and 0T•+• modesare
shown in Figure 1. The toroidal and spheroidalfrequencies, as functionsof •', crossnear 0S•, 0&9, and 0S32. The
resultsfor the multiplet pair 0S•4-0T•5. In model 1066A,
Ato/2rr--• 18/xHz for this pair. In Figure 2a, we plot
singleteigenfrequencies
versus 1000/Q for two cases:(1)
oS•4 -
(o)
6.0
oT•5
6.0 •
5.5
5.5
5.0
5.0
4.5
4.5
4.0
4.0
3.5
5.5
2.210
2.220
2.230
2.210
o•
2.220
2.230
frequency (mHz)
Fig.2. Coupling
between
oS•4andoT]5. (a) No coupling
(plussigns)versus
coupling
(octagons)
amongall nearby
(+ 1 mHz)fundamental
modesdueto rotation,
ellipticity
andattenuation.
(b) No coupling
(plussigns)versus
coupling(octagons)
between
oS•4andoT15dueto rotation,ellipticity,
attenuation,
andaspherical
structure
of Masters
et
al. [1982].
MASTERS
ET AL.:OBSERVATIONS
OFCOUPLEDMODES
oS2•
0
oT2a
(o)
7.0
0
0
-
(b)
'••
7.0-
6.5
6.5-
6.0
6.0-
5.5
5.5-
5.0
5.0
4.5
4.5
4.0
,/.
i
10,289
•o0
I
i
I
2.970 2.980
2.990 5.000 ,3.0'10
2.970 2.980 2.990
i
3.000
I
3.010
frequency (mHz)
Fig. 3. CouplingbetweenoS21and 0T22- Detailsas in Figure2.
no coupling between the two multiplets, and (2) unrestricted coupling, due to rotation, ellipticity, and attenuation, among the fundamental modes 087--0823 and
0T7-0 T24 (all within _+1 mHz of 0S14-0T•s). Only the
singletsof 08•4 and 0T•s enter stronglyin the coupling of
that pair of multiplets. Figure 2a is imperceptiblydifferent
if the basis0&4-0 Tls is used in place of the larger one. In
Figure 2b we compare the uncoupled case to coupling due
to rotation, ellipticity, attenuation, and the If- 2 transition
zone anomalyinferred by Masterset al. [1982] in their
study of fundamental mode peak shifts. Only 08•4 and
0T•s are included in the calculationsfor Figure 2b. The
transition zone model breaks azimuthal symmetry, causing
more generalcouplingand increasingthe effort required to
oSll
solve the matrix eigenvalue problem. Azimuthal symmetry breaking is caused by any asphericalstructure that is
not rotationallysymmetric, e.g., any of the popular, ,,tectonicallyregidnalizedmodels. We have chosenthe transition zone model simply to illustrate the effect of aspherical
structure. In fact, once we can successfullyaccount for
rotation and ellipticity of figure in the coupled multiplets,
the remaining features must be causedby asphericalstruc-
ture including,perhaps,anisotropy. It is the revelationof
aspherical structure that is the ultimate goal of studying
coupling and splitting.
The qualitative behavior for the more complicated
model is similar to that of the model without the Masters
et al. [1982] anomaly. The mean multiplet peak shiftsare
-
oT•2
(o)
+OF
(b)
o
5.0
5.0-
o
o
o
4.5-
o
4.5
4.0
4.0
3.5
3.5
o
o
o
I
I
1.856
I
I
1.860
ooo©
-
+ + + -t+-t+•
3.0
1.852
eo •,•
o
+ + +-••
I
I
1.864
I
I
1.868
3.0
1.852
I
f
1.856
I
I
1.860
.J.•_.
1.854
frequency (mHz)
Fig. 4. Couplingbetween0Sll andoT12. Detailsas in Figure2.
i
1.868
10,290
MASTERSET AL.:OBSERVATIONS
OF COUPLEDMODES
approximate and exact frequency shifts
oSt• multiplet. This is due to reversalsin the interlacing
of the zeroth-orderspheroidaland toroidalsingletreal-part
frequencies caused by the fact that the first-order rotational splittingof 0S• is oppositeto that of 0T•2 (i.e., the
Coriolissplittingparameterfor 0S]• is negative). Relative
spacingof zeroth-ordersinglet eigenfrequencies
is highly
variable in closely spaced multiplet pairs, leading to a
breakdownin the accuracyof (19). To demonstratethis,
we show in Figure 5 mean frequencyshifts of hybrid multiplets, oSt•,relative to the degenerateeigenfrequency
of
the zeroth-ordermultiplets,0&, for model 1066A. All calculations were done with rotation, ellipticity, and attenua-
4
._o
2
E
.E
1
._
•
0
r- --1
$
--2
-3
-4
i
i
i
i
i
i
i
i
i
El
10
12
14
16
18
20
22
24
tion included. The 0S•and 0T•+•dispersionbranchescross
spherical harmonic degree 1•
Fig.5. Meanfrequency
shiftsof oStfhybridmultiplets
relativeto
the unperturbed
0St,multipletsfor model1066A. The effectsof
rotation, ellipticity and attenuation have been computed. Triangles denote results of the full Galerkin calculation. Squares
denote values derived from (19). The basisset used in the calculations consists of all fundamental
modes below 6 mHz.
near f= 11 and f= 19, where we observe "tears"in the
computed frequency shifts. Shifts calculated from (19)
are plottedfor f < 25 as a comparison.As expected,(19)
fails near the "tears."
The results in Figure 5 show clearly the failure of the
diagonalsum rule [Gilbert,1971] when the conditionsfor
its applicationare violated. In the presenceof coupling,
dominated by Coriolis coupling. A similar comparison the generalizationof Woodhouse
[1980] must be used. In
holdstrue for Figure 3, where the frequenciesof 0S2•and the present situation the mean frequency and attenuation
0T22are seen to coupleweakly. The basisfor Figure 3a is of each super multiplet, oStf+oTsl+•, f=8-22,
is
all fundamentalswithin _+1mHz of 0S2• and 0T22. In required. Although there are very many observationsof
Figure 3b the basis is oS21-oT22.Coupling for 0Sll and oSt•modes,thereare very few observations
of 0Ts•modes.
0T]2 is shown in Figure 4. The basisfor Figure 4a is all Until such observations are available we cannot use the
fundamentals within _+1mHz of 0S]•-0T•2 and for generalized diagonal sum rule.
Figure 4b is 0S•l-0T12. 0S•-0T]2 is well known as a
We have demonstratedthat the hybrid multiplets with
stronglycoupledmultiplet pair, with many spheroidaland dominantly spheroidalcharacteristics(St modes) have a
toroidal singletsinterlacedin frequency. Figure 4a makes mean frequency which is shifted from that of a pure
clear that the two typesof zeroth-ordersinglets,however, spheroidal mode. This is illustrated in Figure 6 where
are well separatedin attenuation. 0S]• and 0T•2 couple to observedSt means [Mastersand Gilbert,1983] (see the
form two distinctgroupsof singlets,asidefrom two rela- appendix)and calculated
St means,shownin Figure5, are
tively uncoupledtoroidal singletscorrespondingto 0T12. graphed relative to model 1066A. The deviation of the
These can be thought of as hybrid multiplets, quasi- calculated St means from 1066A is a result of the
toroidal 0Tsl2 and quasi-spheroidaloSt•. The frequency
shiftingis much greaterthan in the earlierexampleand is
Fundamental mode attenuation
7.0
strongly affected by the addition of the Masters et al.
[1982]asphericity.
The overlap in singlet attenuation rates of the two
hybrid multipletsis due to spheroidal-dominant
singletsin
the 0Ts•2 multiplet and toroidal-dominantsinglets in the
6.5
6.0
Computed ond observed frequency shifts
5.5
3-
o 5.0
o
o
o
'-
,_
•
c •t•1
,
ttt•
4.5
4.0
',
3.5
,,
3.0
I
II
-3
,
0
I
I
I
I
I
I
I
I
I
5
10
15
20
25
30
35
40
45
Sphericolhormonicdegree•
Fig. 6. Observed mean frequencies (dots with error bars) and
10
15
20
25
30
35
40
45
Spherical harmonic degree •,
Fig. 7. Observed attenuation (dots with error bars) and calcu-
calculated
meanfrequencies
(triangles)
of oSt[hybridmultipierslatedattenuation
(triangles)
foroSt•modes
andcalculated
attenuarelativeto the unperturbed
0S[multiplets
for model]066A. The tion(continuous
curve)for0Sfmodes
formodel1066A.Thecal-
calculationsare the same as those in Figure 5.
culationsare the same as those in Figure 5.
MASTERSETAL.'OBSERVATIONS
OFCOUPLEDMODES
2.8-
10,291
2.6-
.0
2_.0
3.0
1.o
3.6
316
frequency (rnHz)
frequency (mHz)
•INA / MD IDA 1981'14õ.'õ: 28.' :>0.0
NNA/MD
IDA
1981;145.'5.'28-'20.0
Fig. 8a. Amplitude spectra for fundamental modes on a verti-
cally polarized a½½elerometer.
Unperturbedspectrum (dashed Fig. 8c. Comparison of coupling between nearest neighbors with
line) versusthe spectrumfor modes coupledby rotation, ellipti- (solidline) and without (dashedline as in Figure8b) the asphericity and attenuation(solidline). In the latter, coupling;
amongall cal structure of Masterset al. [1982].
angularorders •< 26 has been computed.
spheroidal-toroidalmultiplet couplingand frequencyrepulsion. Adding the asphericalstructure of Masters et al.
[1982] causesonly very minor changesto the calculatedSt
means. The coupling between 0S]] and 0T12,and 0S]9and
0T20, is so strong that our observedmeans are not just
0T3• and 0S32for a sphericallyaveragedearth should therefore be 6-7/xHz rather than the 0.6/xHz given by 1066A.
The additional separation in frequency is enough to
remove the predicted strong coupling at the crossoverof
the toroidal and spheroidal branches.
Another feature of the St modespredictedby the calcu-
means of St modes but a mixture of Ts and St modes so
lationis the shiftingof the mean Q-• awayfrom that of a
the "tears" are not as pronounced as in the calculation.
The St means for 0S?-•0,0S12-18and 0S20-25are quite well
predicted,though the model could clearlybe improved.
Above f= 26 the model systematically underpredicts
the frequencies,but more importantly, there is a predicted
tear at f-- 32 which is not present in the data. A preliminary investigation of transverse components obtained
from the GDSN indicates that model 1066A systematically
overpredicts the toroidal mode frequencies by about
pure spheroidalmode. This is illustrated in Figure 7. The
Q model used in the calculationis taken from the appendix and predicts the spheroidal mode attenuation shown
by the solid line. The calculated St mode attenuations are
6/xHz at f--• 31. The differencein frequencybetween
2.6-
shownby the triangles,and belowf = 30, their behavioris
clearly reflected in the data. The predicted behavior at
f= 32 is not seen in the data, again suggestingthat the
crossover of the toroidal mode and spheroidal mode
branchesis incorrectly predicted by the model.
While constructingthe Q model in their Table 9, Mas2.8
i
1.O
3.0
36
1.o
2.0
frequency (mHz)
NNA /MD
IDA
3'.6
frequency (rnHz)
1981=145=5:28-'20.0
NNA/MD
IDA
1981:145'5,28:20.0
Fig. 8b. Comparison of coupling between nearest neighbors
Fig. 8d
(dashed line) and all angular orders •< 26 (solid line as in
Figure8a).
includingasphericalstructure(solidline as in Figure8c) and the
unperturbedspectrum(dashedline as in Figure8a).
Comparison of coupling between nearest neighbors,
10,292
MASTERS
ETAL.'OBSERVATIONS
OFCOUPLED
MODES
2.0
structing more accurate, sphericallyaveraged earth models.
Another important consequence of our calculations is
the predicted mode mixing. An example of this is the
appearance, on the spectra of 9ertical component seismic
data, of resonance peaks at frequencies appropriate to
toroidal multiplets. As we have seen, the coupled quasi-
toroidal multiplet oTs• containscomponentsof spheroidal
motion. These components cause the predicted resonance
peak in the vertical component spectrum. The horizontal
component seismic data, after rotation into radial and
transverse components, will show similar, apparently
anomalous peaks. A certain amount of spheroidal motion
in the transverse component and toroidal motion in the
radial component occursindependentof coupling effects,
its amount varying as the inverse of mode angular order f.
0.0 k,
3•6
tO
frequency (mHz)
NNA/MO
IDA
For large f this effect is small, and Coriolis coupling
shouldbe the strongereffect.
We first demonstrate the mode mixing effects with
19B1;145:5.28.'20.0
some synthetic examples. We have used a double couple
Fig. 9a. Comparisonof coupled (solid line) and uncoupled point source located at 48.8øS, 164.4øE and have com(dashedline) amplitudespectra
for a shallow,strike-slip
source.
puted synthetic seismograms for various source depths,
sourcemechanismsand couplingeffectsat the IDA station
ters and Gilbert [1983] realized that the data for angular NNA. Only fundamental mode toroidal and spheroidal
orders 11, 18, and 19 were biasedby couplingeffectsand modes are included in the calculation. Figure 8 illustrates
were not pure spheroidal mode attenuation measurements.
They did not realize that all the measurements for
f = 10-21 are significantlybiased, and this led to a model
with a lower mantle which is too highly attenuating.
Further modeling of high-quality attenuation measurements must incorporatethe couplingeffects, and a prelim-
inaryresultis presentedin the appendix.
The same point is also true for the frequencydata. The
patern of frequencyshiftsin Figure6 wasfirst notedby
Silverand Jordan[1981] and is presentin the data usedto
construct current earth models. These data obviously cannot be regarded as belonging to pure spheroidal modes,
the results for a 10-km-deep thrust mechanism dipping at
45ø and striking north. The spectrahave been computed
for a 40-hour record length which has been tapered using
a Hanning window. Figure 8a is a comparison of a spectrum for no coupling with that of coupling due to the
Coriolis force and ellipticity. Ts modes are clearly visible
and, as expected, are located approximately at the toroidal
mode degenerate frequencies. It should be noted that
tapering is essential to reduce sideband leakage in the
spectrum which otherwise masks the Ts modes. This fact
might explain why Ts modes have not previously been
and their errors are now sufficientlysmall that a bias of 10
or more standarddeviationscould result if mode coupling
effects were not considered.
o
In summary, we believe that these results demonstrate
the importance of considering mode coupling when con-
o
if
022
o 2o
018
016
'o
0.14
._
•.
012
o•o
=
0.08
._
0.06
0.04
0.02
0.00
1.6
1.8
2.0
2 2
2 4
2 6
2 8
3 0
3 2
frequency (rnHz)
KIP
0.0
frequency (mHz)
NNA/MD
IDA
1981;145.'5,28:20.0
Fig. 9b. As in Figure 9a for a deep, thrust source.
1981'145.5
29'50.0
Fig. 10. Observed spectrum showing Ts modes on a vertically
polarized IDA (La Coste-Romberg)accelerometer.A 35-hour
record has been Hanning-tapered to produce the spectrum. The
station/sourcefor this spectrumis KIP/New Zealand 1981.
MASTERS
ETAL.'OBSERVATIONS
OFCOUPLEDMODES
observed in real data. A long record length is also
required to give the frequency resolution necessary to
separate the Ts and St modes. It should be emphasized
that this has the effect of suppressingthe amplitude of the
Ts modes relative to the St modes, as they generally have
a higher attenuation rate. Deviations in the coupled spectrum from the uncoupled spectrum are quite pronounced
10,293
0.0008
0.0006
throughout
thisfrequencyband(1.0-3.5 mHz).
This calculationis complete in that coupling between all
singlets(t •< 26) of the same azimuthal order has been
included.
If
we restrict the calculation to coupling
0.0004
betweennearestneighbor0S!,0Tt+l modes, we have the
result in Figure 8b. This demonstratesthat basicallyall
the coupling in this frequency band is due to nearest
neighbors.
An additional aspherical structure greatly increases the
labor involved in solving the coupling problem so we
demonstratethe effect by including only nearest neighbor
0S!,0Tt+•modes.The t•= 2 transition
zonemodelof Mas-
0.0002
0'00005 16
1.7
18
19 20 2 1 22 23 24
26
frequency (mHz)
GRFO
1981:
145:5:29:50.0
ters et al. [1982] has been employed, and the result is
Fig. 12. As in Figure 10 for a vertically polarized SRO
shown in Figure 8c. The spectrum including coupling accelerometer:
GRFO/New
Zealand 1981.
effects due to rotation, ellipticity, and asphericalstructure
is comparedwith the spectrumincludingcouplingeffects
due to ellipticity and rotation between nearest neighbors.
The two spectra are very similar except that there appears and toroidal modes was not included (J. H. Woodhouse,
to be some broadeningof the hybrid multipletsdue to the personalcommunication,1982). The amplitudevariations
asphericalstructure. This effect makesthe Ts modesless
clear when compared with the uncoupled spectrum as
illustratedin Figure 8d.
We think that these calculations show that the dominant
couplingeffect in this frequencyband is the Coriolisforce
actingon nearestneighbor0S•, 0T•+• modes. The amplitude differences between coupled and uncoupled multiplets are similar to those observed between real data and
synthetics computed using a spherically symmetric earth
model [Riedeselet al., 1981] and certainly cannot be
regardedas small. Large-scalecouplingexperimentshave
been made in which Coriolis coupling between spheroidal
o
o
o
o
o
o
o
o
o
o
o
016
0.14
012
•
010
._
in these restricted calculations did not exceed 10%.
We have investigated the effect of varying the source
parametersand illustrate some of the results in Figure 9.
Figure 9a showsthe effectof a strike-slipsourcelocatedat
10-kin depth with a northerly strike and a vertical fault
plane. The toroidal modes are better excited relative to
the spheroidal modes for this kind of source; consequently, the Ts modes are more pronounced. The cou-
pling calculationincludesthe effectsdue to rotation and
ellipticity. The differences between the coupled and
uncoupled spectra are again quite pronounced over the
whole frequency interval.
Figure 9b gives a comparison for a deep thrust event
located at 600-kin depth. The coupling calculation
includes only the effects due to rotation and ellipticity.
We had expectedthe differencesbetweenthe coupledand
uncoupled spectra to be less for deep earthquakesas the
toroidal excitations are very small. This is not the case,
and the coupling again gives Ts modes and strong variations in the amplitudes of the St modes.
If we inspect vertical component spectraof earthquakes
which are rich in low-frequency toroidal mode energy, the
syntheticcalculationssuggestthat the Ts modes should be
clearly visible. The station geometry used in these exam-
E
0 08
ples is that of an event which occurredon May 25, 1981.
•
006
Data from this event have provided the clearest observations of Ts modes to date. Figures 10-13 give examples
from different kinds of vertical component instruments
._
_
recording this event. All spectra are for 35-hour time
0.04
Fig. 11. As in Figure10 for the UCLA (La Coste-Romberg)
sta-
intervals and have been tapered using the Hanning window. Ts modes are clearly visible at nearly all stations and
are easily identified on IDA and GDSN instruments. It is
easy to show that Ts modes behave like quasi-toroidal
modes with a higher attenuation rate than the St modes
by computingspectrafor different start times. As the start
time is delayed relative to the origin time of the event, the
tion (South Pole)/source: SPA/New Zealand 1981.
Ts modesdecayrelativeto the St modes(Figure 14). We
002
0.00
1.8
2 0
2 2
2.4
2 6
2 8
3 0
32
frequency (mHz)
SPA
1981-' 145:
5: 29:50.0
10,294
MASTERSET AL.' OBSERVATIONS
OFCOUPLEDMODES
o
%
o
..o
% •
•r io•=• • • io
0.0018
0.0016
0.0014
•
0.0012
"•
0.0010
•
0.0008
--
0.0006
spheroidal modes so Ts modes are weak on both the vertical and radial components. Conversely, the St modes are
relatively strong on the transverse component. As most
large events are of thrust or normal type, like the Tonga
event, it may be difficult to get measurements of Ts
modes from transverse components because of the proximity of the strong St modes.
4.
._
At frequencies below 3.0 mHz, fundamental spheroidal
and toroidal modes are coupled strongly and are observed
as hybrid multiplets which are shifted in both mean frequency and attenuation rate from the uncoupled values.
Except for very strong coupling, the hybrid multiplets are
._
0.0004
predominantly spheroidal (St modes) or toroidal (Ts
0.0002
i
0.0000
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
frequency (mHz)
KONO
Fig. 13. As
accelerometer:
1981=145:5=29=50.0
in Figure 10 for a vertically polarized ASRO
KONO/New
Zealand 1981.
again emphasize that the procedures we have used to
enhancefrequencyresolutionin the spectra(long record
length and tapering)suppressthe apparentamplitudesof
the
DISCUSSION
Ts modes relative
to the St modes because of the
modes) in character. We have demonstrated that the
shifts in the mean frequenciesand attenuation rates of St
modescan be predictedby consideringthe couplingeffects
of rotation, attenuation, and ellipticityof figure. Of these,
the first is dominant and the second is very important.
We have not found it necessaryto include anisotropy in
the calculations. If anisotropy is confined to the upper
mantle, then coupling effects may become noticeableat
frequencieshigher than 3.0 mHz as the energy distribution
of the modes becomes more concentrated in this region.
(It shouldalso be rememberedthat a transverselyisotropic
model, such as PREM, would not give any coupling. Cou-
difference in attenuation rate. In some of the examples
pling occursonly when sphericalsymmetryis broken.)
(e.g., KIP) the initial amplitudesof the Ts modes are
Another factor which arguesstronglyfor the dominanceof
Coriolis coupling is that the coupling appears to be
comparablewith the initial amplitudesof the St modes. It
is clear that unlesswe model the couplingeffectswe will
not be able to perform detailed modeling of the source in
this frequency band.
confinedto nearestneighborsin the data, i.e., oS•-oT•+l
These observations establish Ts modes as a reality and
have encountered when one rotates horizontal component
seismogramsto get a transverseand a radial component.
•(
i/
I
0.12
I
I
I
I
I
I
I
•
I
0.10
On a true (transverse,radial) componentwithout coupling
0.08
factor of f relative to the other component. Thus both
componentsshould show both mode types for low f but
0.06
I
I
I
I
I
I
I
I
I
0.04
I
I
I
I
•
i i
0.02
I
I
o. oo
•.8 1:
•o
II
I
Ii
=
0.06
I
i
II
i
,fi
'
I
ß
,
I I
0.05 I
i
I
the instruments are properly calibrated. However, above
0.04
•
I
1.7 mHz there are severalexamplesof 0S•-0 T•+i coupling.
0.03
i
,,
I
Ill
•
•
I
are no 0S•modes. We think that theseresultsmean that
This pattern is repeated for ANMO/Tonga 1977 whose
radial and transverse spectra are shown in Figure 16.
Again, below 1.7 mHz the radial spectrumis dominantly
' ,
I
I
the (,S•, , T0 modesshouldbe reducedapproximately
a
not for f >> 1. In Figure 15 we have the spectrumof the
radial component of NWAO/Tonga 1977. With the
exceptionof the low f, iT2 mode all modesare, S• modes.
There are no 0T• modes except the strongly coupled
0T]2-0S]] pair. Also in Figure 15 is the spectrumof the
transverse component of NWAO/Tonga 1977. Below
1.7 mHz the only ,S• mode is 2S6,stronglyexcitedon the
radial componentand down a factor of 3.4 here, and there
hr
0.14
suggesta solution for another observationalproblem we
•
' II
i•
I
I
s
I
'
I
I
I
0.02
O.Oi
, S• and the transversespectrumis dominantly,T•, implying good calibration. Again, above 1.7 mHz there is evi-
dence of oS•-oT•+• couplingwith some prominent St
modes on the transverse component. This is a common
feature of spectra from 1.7 to 2.7 mHz on rotated components of both SRO and ASRO horizontals. The Tonga
event did not excite toroidal modes as strongly as
0'001.8
2.0
21
.2
2.4
2.6
frequency (mHz)
Fig. 14. As in Figure 10 for CMO/New Zealand 1981. The
upper spectrumcorrespondsto a start time of 0 hours, and the
lower 5 hours. The time lapse spectra show that Ts modes
attenuate more rapidly than St modes.
MASTERS
ETAL.:OBSERVATIONS
OFCOUPLED
MODES
o
o
o
o
o
10,295
o
2.8-
0.0
.•
.
1.5
i
:•.0
:•.5
frequency (mhz)
NWAO/R
SRO 1977:173:12:15:10.0
Fig.]5. Spectra
of'40-hourHarming-tapered
records
for NWAO/Tonga]9??. Radial(dashed
line) andtransverse
(solid
line)spectra
showing
oS•,-o
T•,+]
coupling.
or 0S•-0T•-]. It seemsthatproximity
in angularorder,'f, couplingas the dominantfactorratherthan anisotropyor
as well as in complexfrequencyis requiredto produce aspherical structure.
observablecoupling.Also the type of mode, S or T, is
In our experiments, aspherical structure does cause
important.For example0T?and0S?differin frequencyby noticeablechangesin the calculatedspectraof very
only 8.8 •Hz and have Q valuesof about200 and 400, stronglycoupled modes. In principle, we could use this
respectively,but are not observedto be coupled. These feature to distinguishbetween competingmodels of
factorssuggestthat we are correctin identifyingCoriolis asphericalstructure. In practice, such an experiment
o o
0.0
1.2
o
o
o
o
o
o o
'
1ß5
"
2•.0
o o
'
o
o
o
•
o
o
o
•
2.5
frequency (mHz)
ANMO/R
SRO
1977:175:12.'15:10.0
Fig.16. Asin Figure15forANMO/Tonga
1977.Several
St modes
appear
onthetransverse
component.
10,296
MASTERS
ETAL.:
OBSERVATIONS
OFCOUPLED
MODES
TABLE A1. A SimpleO Model with Bulk O Constrained
to be
at Least 50 x Shear O
Model
couplingis strong. The shiftsin peakfrequencyreported
by Silverand Jordan[1981]and Masterset al. [1982]
represent observationsof singletlike multiplet peaks.
Parameters
ShellRadii,km
ShearQ
Bulk Q
0-- 1229
3484-- 5700
5700--5950
5950--6371
3140
332
277
118
157,000
16,600
13,850
5,895
These peak shifts exhibit a clear dependenceon source-
receiver path and led to a preliminaryinversionfor
aspherical
structureusingthe asymptotic
theoryfor uncoupled modesderivedby Jordan[1978]. The meanpeak
shift of an uncoupledmode when averagedover all
source-receiver orientations would be zero, a consequence
of the diagonalsum rule. This is not true for coupled
hybrid
modesas demonstrated
in Figure6. At frequenwouldbe premature.A veryprecisemodelof the sphericies
below
about
3.0
mHz
we
observe
St modes, and we
cally averagedstructureis requiredbecausethe very
strongcoupling
is sensitive
to the exactrelativelocations would expectthe peak shiftsof thesemodesto showa
on source-receiver
locationfrom
of the fundamental spheroidal and toroidal mode differentdependence
uncoupled
spheroidal
modes.Thisfeaturecanbe seenin
branches. Such a model is not yet available.
et al. [1982],wherethe spherical
harThe differencesbetween the computed spectrafor a Figure2 of Masters
spherical
earthandformorerealistic
(though
verysimple) monic coefficientsof the peak shift patternsare clearly
bythecoupling
effectsfor oSt8-oSt2o.
It is obvimodels demonstrate the potential for bias in source perturbed
to extendthe theoryfor interpreting
peak
mechanismretrieval if couplingis ignored. It may be pos- ouslydesirable
sible to misinterpretthe couplingsignalas a complex shifts to include the coupledhybrid modes.
In summary, the frequencyrepulsion,attenuation
source mechanism. We are currently investigatingthis
andamplitude
variation
in thehybridmultiplets
matter,andwesuspect
thatcoupling
mayexplain
someof averaging,
the difficultieswe have experiencedin retrieving source are observedfeaturesthat must be taken into accountin
studiesof modelsof density,elasticstructure,and anelasmechanisms,
particularlythoseof shallowevents.
A reexaminationof the methods used to infer aspheri- tic structure and in studiesof earthquakesourcemechan-
cal structureis alsorequiredin the frequencybandswhere isms. The computationalproblemat low frequenciesis
TABLE
A2.
Fundamental
Spheroidal
ModeFrequency
andAttenuation
Values
Standard
Degree
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
Frequency, Deviation,
mHz
/xHz
1000/Q
1.41240
1.57675
1.72363
1.86280
1.99093
2.11320
2.23109
2.34620
2.45833
2.56722
2.67330
2.77602
2.87670
2.97614
3.07388
3.17032
3.26520
3.35880
3.45135
3.54305
3.63425
3.72440
3.81464
3.90480
3.99416
4.08296
4.17130
4.26125
4.35090
4.44050
4.52984
4.61830
4.70740
4.79795
4.88795
4.97702
0.15
0.15
0.15
0.30
0.15
0.15
0.15
0.20
0.15
0.15
0.30
0.35
0.35
0.15
0.15
0.15
0.25
0.15
0.20
0.20
0.25
0.15
0.25
0.30
0.25
0.20
0.30
0.35
0.30
0.40
0.30
0.50
0.40
0.60
0.75
0.30
*Total number of observations is 8302.
2.72
2.91
3.00
4.14
3.18
3.23
3.30
3.14
3.40
3.74
4.10
4.37
3.93
3.84
3.93
4.00
4.09
4.29
4.30
4.53
4.73
4.62
4.76
5.12
5.15
5.48
5.25
5.61
5.56
5.57
5.77
6.03
6.10
6.23
6.27
6.32
Standard
No.of
Deviation Observations*
0.10
0.07
0.08
0.18
0.06
0.03
0.05
0.04
0.04
0.05
0.05
0.06
0.06
0.04
0.04
0.03
0.06
0.05
0.05
0.05
0.05
0.05
0.07
0.08
0.08
0.06
0.09
0.11
0.07
0.07
0.06
0.13
0.14
0.12
0.18
0.15
118
186
188
63
172
261
202
246
261
219
236
257
237
361
331
446
278
300
345
328
350
390
274
262
239
337
216
153
209
194
213
108
111
98
56
57
MASTERS
ETAL.:OBSERVATIONS
OFCOUPLEDMODES
10,297
made more complicatedby multiplet coupling but leads to
a more refined analysisof the complex frequency data and
complex amplitude data.
That strong coupling is very sensitive to structure is a
severity that must be overcome in the analysis. In the
long run it may prove to be a blessingin disguisebecause
the details of strong coupling, if they can be observed,
should provide very strong constraintson some features of
Benioff, H., F. Press, and S. W. Smith, Excitation of the free
oscillations of the earth by earthquakes, J. Geophys.Res., 66,
605-619, 1961.
Dahlen, F. A., The normal modes of a rotating, elliptical Earth,
Geophys.
J. R. Astron.Soc., 16, 329-367, 1968.
Dahlen, F. A., The normal modes of a rotating, ellipticalearth, II,
Near resonancemultiplet coupling, Geophys.J. R. Astron. Soc.,
18, 397-436, 1969.
Dahlen, F. A., The effect of data windows on the estimation of
free oscillation parameters, Geophys.J. R. Astron. Soc., 69,
models of the earth.
537-549, 1982.
Dahlen, F. A., and R. V. Sailor, Rotational and elliptical splitting
of the free oscillationsof the earth, Geophys.J. R. Astron.Soc.,
APPENDIX
58, 609-623, 1979.
Dziewonski, A., and D. Anderson, Preliminary reference earth
The model of attenuation used in this paper has been
model, Phys.EarthPlanet.Inter., 25, 297-356, 1981.
derived from 57 observed mean attenuation data. The
Finlayson, B. A., The Method of WeightedResidualsand Variational
Principles,Academic, New York, 1972.
data have been adjustedto remove the effect of coupling.
Garbow,
B., J. Boyle, J. Dongarra, and C. Moler, Matrix EigensysLet q -- Q-1 anddefine
tem Routines--EISPACK Guide Extension, Lecture Notes in
ComputerScience,Springer-Verlag,New York, 1977.
q2u calculatedvalue of q with no coupling;
Gilbert, F., The diagonalsum rule and averagedeigenfrequencies,
Geophys.
J. R. Astron.Soc.,23, 125-128, 1971.
qc calculatedvalue of q in the presenceof rotation,
Gilbert, F., and A. Dziewonski, An applicationof normal mode
ellipticity of figure and asphericalstructure;
theory to the retrieval of structural parameters and source
qo observed value of q;
mechanismsfrom seismic spectra, Philos. Trans. R. Soc. London
q,• adjustedobservationwith couplingeffectsremoved.
Ser. A, 2 78, 187-269, 1975.
Jordan, T. H., A procedurefor estimatinglateral variations from
low-frequencyeigenspectradata, Geophys.J. R. Astron.Soc., 52,
Then, for a good model of attenuation, by definition
441-455, 1978.
qo- qc -- qA - q•t or
Kanamori, H, and D. L. Anderson, Importance of physicaldispersion in surface wave and free oscillationsproblems: review, Rev.
qA= qo - (q½-q•t)
(A1)
Geophys.
SpacePhys.,15, 105-112, 1977.
We have applied(A1) to the observations
of 0S• for Kantorovich, L. V. and V. I. Krylov, ApproximateMethods of
HigherAnalysis,P. Noordhoff, Groningen, 1958.
f= 8-46. Both qc and q2u were calculated from the
Kawakatsu, H., and R. J. Geller, A new iterative method for
model in Table 9 of Masters and Gilbert [1983]. The
finding the normal modes of a laterally heterogeneous body,
remaining data were not adjustedfor couplingeffects. The
Geophys.
Res. Lett., 8, 1195-1197, 1981.
0S•data of Mastersand Gilbert[1983], derivedfrom 557 Liu, H.-P., D. L. Anderson, and H. Kanamori, Velocity dispersion
due to anelasticity;implicationsfor seismologyand mantle comIDA records, have been supplemented. The new 0S•
position,Geophys.
J. R. Astron.Soc., 47, 41-58, 1976.
dataset has been derived from 950 IDA records and 250
Luh,
P.
C.,
Free
oscillations
of the laterallyinhomogeneousEarth:
GDSN vertical records.
Quasi degeneratemultiplet coupling, Geophys.J. R. Astron.Soc.,
Using the adjusted dataset, we have derived the model
32, 187-202, 1973.
listed in Table A1. The details of the inversion procedure Luh, P. C., The normal modes of the rotating self gravitating
inhomogeneousearth, Geophys.J. R. Astron.Soc., 38, 187-224,
are given by Mastersand Gilbert[1983]. We regard the
ß
model in Table A1 to be an interim model, derived primarily to provide a better fit to the observed data with
1974.
MacDonald, G. J. F., and N. F. Ness, A study of the free oscilla-
tionsof the earth, J. Geophys.
Res., 66, 1865-1911, 1961.
couplingeffectsmodelledaccordingto (A1). Eventually, Madariaga, R. I., Free oscillationsof the laterally heterogeneous
we should be able to model the observed attenuation
of
individualsinglets,and the expedientof using(A1) canbe
surpassed.The new 0S•datasetis givenin Table A2.
Acknowledgments.
This work was supportedby the National
Science Foundation grants EAR 80-25261 and EAR 81-21866.
We are very grateful to D.C. Agnew and J. Berger for maintaining the consistentlyvery high qualityof data from the IDA network, and to F. A. Dahlen for constructivecomments.
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(ReceivedApril 8, 1983;
revisedAugust 19, 1983;
acceptedAugust19, 1983.)